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Corollary 5.7. Let \( C \) be a closed convex set in \( {\mathbb{R}}^{d} \). Then the sets \( \operatorname{ri}F \), where \( F \in \mathcal{F}\left( C\right) \smallsetminus \{ \varnothing \} \), form a partition of \( C \). | Proof. The statement amounts to saying that for each \( x \in C \) there is a unique face \( F \) of \( C \) such that \( x \in \operatorname{ri}F \). However, Theorem 5.6 gives such a unique face, namely, the smallest face of \( C \) containing \( x \). | Yes |
Theorem 5.8. Let \( F \) be a facet of a closed convex set \( C \) in \( {\mathbb{R}}^{d} \) . Then \( F \) is an exposed face. | Proof. By the definition of a facet we necessarily have \( \dim F \geq 0 \), whence by Theorem 3.1 there is a point \( x \in \mathrm{{ri}}F \) . Then, by Theorem 5.6, \( F \) is the smallest face of \( C \) containing \( x \) . On the other hand, Theorem 4.3 shows that there is an exposed face \( G \) of \( C \) such t... | Yes |
Theorem 5.10. Let \( C \) be a compact convex set in \( {\mathbb{R}}^{d} \), and let \( M \) be a subset of \( C \) . Then the following two conditions are equivalent:\n\n(a) \( C = \operatorname{conv}M \) .\n\n(b) ext \( C \subset M \) .\n\nIn particular,\n\n(c) \( C = \operatorname{conv}\left( {\operatorname{ext}C}\r... | Proof. Suppose that there is an extreme point \( x \) of \( C \) which is not in \( M \) . Then \( M \) is a subset of \( C \smallsetminus \{ x\} \), and since \( C \smallsetminus \{ x\} \) is convex by the definition of an extreme point, it follows that conv \( M \) is also a subset of \( C \smallsetminus \{ x\} \) . ... | Yes |
Corollary 5.11. Let \( C \) be a compact convex set in \( {\mathbb{R}}^{d} \) with \( \dim C = n \) . Then each point of \( C \) is a convex combination of at most \( n + 1 \) extreme points of \( C \) . | Proof. Combine Theorem 5.10(c) and Corollary 2.4. | No |
Theorem 6.1. For any subset \( M \) of \( {\mathbb{R}}^{d} \) one has:\n\n(a) If \( M \) is bounded, then \( o \) is an interior point of \( {M}^{ \circ } \) .\n\n(b) If \( o \) is an interior point of \( M \), then \( {M}^{ \circ } \) is bounded. | Proof. For \( z \in {\mathbb{R}}^{d} \) and \( r > 0 \) we denote by \( B\left( {z, r}\right) \) the closed ball centred at \( z \) with radius \( r \), i.e.\n\n\[ B\left( {z, r}\right) \mathrel{\text{:=}} \left\{ {x \in {\mathbb{R}}^{d} \mid \parallel x - z\parallel \leq r}\right\} .\n\]\n\nHere \( \parallel \cdot \pa... | Yes |
Theorem 6.2. For any subset \( M \) of \( {\mathbb{R}}^{d} \) we have\n\n\[ \n{M}^{\circ \circ } = \operatorname{clconv}\left( {\{ o\} \cup M}\right) ,\n\]\n\ni.e. \( {M}^{\circ \circ } \) is the smallest closed convex set containing o and \( M \) . | Proof. We have\n\n\[ \n{M}^{ \circ }{}^{ \circ } = \mathop{\bigcap }\limits_{{y \in {M}^{ \circ }}}K\left( {y,1}\right) = \mathop{\bigcap }\limits_{{M \subset K\left( {y,1}\right) }}K\left( {y,1}\right) ,\n\]\n\n(5)\n\ncf. (1) and (2). This formula immediately implies that \( {M}^{\circ \circ } \) is a closed convex se... | Yes |
Theorem 6.4. For any \( y \in {\mathbb{R}}^{d} \), the following two conditions are equivalent:\n\n(a) \( H\left( {y,1}\right) \) is a supporting hyperplane of \( C \) .\n\n(b) \( y \in \mathrm{{bd}}D \) . | Proof. If (a) holds, then \( y \in D \) and\n\n\[ \mathop{\sup }\limits_{{x \in C}}\langle x, y\rangle = 1 \]\n\n(7)\n\n(Actually, the supremum is a maximum.) If we had \( y \in \operatorname{int}D \), then we would also have \( {\lambda y} \in D \) for a certain \( \lambda > 1 \) . Since \( D \) is the polar of \( C \... | Yes |
For any \( x, y \in {\mathbb{R}}^{d} \), the following four conditions are equivalent:\n\n(a) \( H\left( {y,1}\right) \) is a supporting hyperplane of \( C \) at \( x \) .\n\n(b) \( H\left( {x,1}\right) \) is a supporting hyperplane of \( D \) at \( y \) .\n\n(c) \( \langle x, y\rangle = 1, x \in \mathrm{{bd}}C, y \in ... | Proof. The equivalence (a) \( \Leftrightarrow \) (c) follows immediately from Theorem 6.4,(a) \( \Leftrightarrow \) (b). The equivalence (b) \( \Leftrightarrow \) (c) then follows by symmetry, or from Theorem 6.4,(c) \( \Leftrightarrow \) (d). It is trivial that (c) \( \Rightarrow \) (d). We shall complete the proof by... | Yes |
Theorem 6.6. Let \( F \) be a proper exposed face of \( C \). Then \( {F}^{\Delta } \) is a proper exposed face of D. Similarly for a proper exposed face \( G \) of \( D \). | Proof. By definition,\n\n\[ \n{F}^{ \vartriangle } = \mathop{\bigcap }\limits_{{x \in F}}D \cap H\left( {x,1}\right) \n\] \n\nWhen \( F \) is proper, then each \( x \in F \) is in bd \( C \), whence \( H\left( {x,1}\right) \) is a supporting hyperplane of \( D \), cf. Theorem 6.4,(d) \( \Rightarrow \) (c). Therefore, e... | Yes |
Theorem 6.7. Let \( F \) be a proper exposed face of \( C \) . Then \( {F}^{\circ \vartriangle } = F \) . Similarly for a proper exposed face \( G \) of \( D \) . | Proof. By definition,\n\n\[ \n{F}^{\vartriangle \vartriangle } = \mathop{\bigcap }\limits_{{y \in {F}^{ \blacktriangle }}}C \cap H\left( {y,1}\right)\n\]\n\nBut since \( y \) is in \( {F}^{\Delta } \) if and only if \( H\left( {y,1}\right) \) is a supporting hyperplane of \( C \) with \( F \subset H\left( {y,1}\right) ... | Yes |
Theorem 6.10. Let \( F \) and \( G \) be a pair of mutually conjugate faces of \( C \) and \( D \) , respectively. Then\n\n\[ \n\\dim F + \\dim G \\leq d - 1.\n\] | Proof. The conjugate face of the improper exposed face \( \\varnothing \) of \( C \) is the improper exposed face \( D \) of \( D \) . Similarly, the conjugate face of the improper exposed face \( C \) of \( C \) is the improper exposed face \( \\varnothing \) of \( D \) . Since \( \\dim \\varnothing = - 1 \) , \( \\di... | Yes |
Theorem 7.1. Let \( P \) be a non-empty subset of \( {\mathbb{R}}^{d} \). Then the following two conditions are equivalent:\n\n(a) \( P \) is a polytope.\n\n(b) \( P \) is a compact convex set with a finite number of extreme points. | Proof. When \( P \) is a polytope, say \( P = \operatorname{conv}\left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \), then \( P \) is compact by Corollary 2.9. Next, Theorem 5.10,(a) \( \Rightarrow \) (b) shows that ext \( P \) is a subset of \( \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \), and hence is a finite set. The conv... | Yes |
Theorem 7.2. Let \( P \) be a polytope in \( {\mathbb{R}}^{d} \), and let \( \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \) be a finite subset of \( P \) . Then the following two conditions are equivalent:\n\n(a) \( P = \operatorname{conv}\left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \).\n\n(b) ext \( P \subset \left\{ {{x}_... | Proof. Noting that polytopes are compact, the statement follows immediately from Theorem 5.10. | No |
Theorem 7.3. Let \( P \) be a polytope in \( {\mathbb{R}}^{d} \), and let \( F \) be a proper face of \( P \) . Then \( F \) is also a polytope, and ext \( F = F \cap \) ext \( P \) . | Proof. We begin by noting that \( P \) and \( F \) are compact, cf. Theorem 7.1,(a) \( \Rightarrow \) (b) and Theorem 5.1. Now, Theorem 5.2 shows that the extreme points of \( F \) are just those extreme points (vertices) of \( P \) which are in \( F \), i.e. ext \( F = F \cap \) ext \( P \) . Since ext \( P \) is a fi... | Yes |
Corollary 7.4. Let \( P \) be a polytope in \( {\mathbb{R}}^{d} \). Then the number of faces of \( P \) is finite. | Proof. The number of extreme points of \( P \) is finite by Theorem 7.2,(a) \( \Rightarrow \) (b). Each face of \( P \) is the convex hull of extreme points of \( P \) by Theorem 7.3 and Theorem 7.2(c). Therefore, the number of faces is finite. | Yes |
Theorem 7.8. Let \( P \) be a bipyramid in \( {\mathbb{R}}^{d} \) with basis \( Q \) and apices \( {x}_{0} \) and \( {x}_{1} \) . Then the following holds:\n\n(a) \( P \) is a polytope with ext \( P = \left( {\text{ext}Q}\right) \cup \left\{ {{x}_{0},{x}_{1}}\right\} \) .\n\n(b) A subset \( F \) of \( P \) with \( {x}_... | Proof. The proof follows the same lines as the proof of Theorem 7.7. The details are left to the reader. | No |
Theorem 8.1. Let \( Q \) be a polyhedral set in \( {\mathbb{R}}^{d} \) with \( \dim Q = d \) and \( Q \neq {\mathbb{R}}^{d} \). Let\n\n\[ Q = \mathop{\bigcap }\limits_{{i = 1}}^{n}K\left( {{x}_{i},{\alpha }_{i}}\right) \]\n\nbe a representation of \( Q \) with \( n > 1 \), where each \( K\left( {{x}_{i},{\alpha }_{i}}\... | Proof. For \( j = 1,\ldots, n \), we let\n\n\[ {M}_{j} \mathrel{\text{:=}} \mathop{\bigcap }\limits_{\substack{{i = 1} \\ {i \neq j} }}^{n}K\left( {{x}_{i},{\alpha }_{i}}\right) \]\n\nThen \( Q = K\left( {{x}_{j},{\alpha }_{j}}\right) \cap {M}_{j} \) for each \( j \), and since \( \dim Q = d \) by assumption, we see th... | Yes |
Theorem 8.3. Let \( F \) be a proper face of a polyhedral set \( Q \) in \( {\mathbb{R}}^{d} \) . Then there is a facet \( G \) of \( Q \) containing \( F \) . | Proof. We may assume that \( \dim Q = d \) . Choose an irreducible representation\n\n\[ Q = \mathop{\bigcap }\limits_{{i = 1}}^{n}K\left( {{x}_{i},{\alpha }_{i}}\right) \]\n\nLet \( x \) be a relative interior point of \( F \) . By Theorem 8.2(a), there is \( j \) such that\n\n\[ x \in H\left( {{x}_{j},{\alpha }_{j}}\r... | Yes |
Corollary 8.4. Let \( Q \) be a polyhedral set in \( {\mathbb{R}}^{d} \). Then every face of \( Q \) is also a polyhedral set. | Proof. We need only prove the statement for proper faces of \( Q \). Theorem 8.3 shows that any proper face of \( Q \) is a face of a facet of \( Q \). Facets of \( Q \), however, are polyhedral sets by Theorem 8.2(b). The statement then follows by induction on the dimension. | Yes |
Corollary 8.5. Let \( Q \) be a polyhedral set in \( {\mathbb{R}}^{d} \). Then the number of faces of \( Q \) is finite. | Proof. The number of facets of a polyhedral set \( Q \) is finite, cf. Theorem 8.2(b). Each proper face of \( Q \) is a face of a facet of \( Q \) by Theorem 8.3. The statement then follows by induction on the dimension. | No |
Corollary 8.6. Let \( Q \) be a polyhedral set in \( {\mathbb{R}}^{d} \) with \( \dim Q = d \) . Let \( {F}_{j} \) and \( {F}_{k} \) be faces of \( Q \) with\n\n\[ \n{F}_{j} \subset {F}_{k} \n\]\n\nand\n\n\[ \n\dim {F}_{j} = j,\;\dim {F}_{k} = k, \n\]\n\nwhere\n\n\[ \n0 \leq j < j + 1 \leq k - 1 < k \leq d. \n\]\n\nThe... | Proof. By Theorem 5.2, \( {F}_{j} \) is a proper face of \( {F}_{k} \) . And by Corollary 8.4, \( {F}_{k} \) is polyhedral. Theorem 8.3 then ensures the existence of a facet \( {F}_{k - 1} \) of \( {F}_{k} \) with \( {F}_{j} \subset {F}_{k - 1} \) . If \( j = k - 2 \), we have the desired conclusion. If \( j < k - 2 \)... | Yes |
Corollary 8.7. Let \( Q \) be a non-empty bounded polyhedral set in \( {\mathbb{R}}^{d} \). Then \( Q \) is a polytope. | Proof. By assumption, \( Q \) is a compact convex set. By Corollary 8.5, ext \( Q \) is a finite set. The statement then follows from Theorem 7.1,(b) \( \Rightarrow \) (a). | Yes |
Theorem 9.1. Let \( {x}_{1},\ldots ,{x}_{n} \), where \( n \geq 1 \), be distinct points of \( {\mathbb{R}}^{d} \), and let\n\n\[ P \mathrel{\text{:=}} \operatorname{conv}\left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \]\n\n\[ Q \mathrel{\text{:=}} \mathop{\bigcap }\limits_{{i = 1}}^{n}K\left( {{x}_{i},1}\right) \]\n\nThen ... | Proof. (a) Formula (1) of Section 6 shows that\n\n\[ {\left\{ {x}_{1},\ldots ,{x}_{n}\right\} }^{ \circ } = Q \]\n\n(1)\n\nand formula (2) of Section 6 shows that\n\n\[ {\left\{ {x}_{1},\ldots ,{x}_{n}\right\} }^{ \circ } = {\left( \operatorname{conv}\left\{ {x}_{1},\ldots ,{x}_{n}\right\} \right) }^{ \circ } \]\n\n(2)... | Yes |
Theorem 9.2. A non-empty subset \( P \) of \( {\mathbb{R}}^{d} \) is a polytope if and only if it is a bounded polyhedral set. | Proof. We have already proved the \ | No |
Corollary 9.3. Let \( {P}_{1} \) and \( {P}_{2} \) be polytopes in \( {\mathbb{R}}^{d} \) such that \( {P}_{1} \cap {P}_{2} \neq \varnothing \) . Then \( {P}_{1} \cap {P}_{2} \) is also a polytope. | Proof. The intersection of any two polyhedral sets in \( {\mathbb{R}}^{d} \) is polyhedral. The statement then follows from Theorem 9.2. | Yes |
Corollary 9.4. Let \( P \) be a polytope in \( {\mathbb{R}}^{d} \), and let \( A \) be an affine subspace of \( {\mathbb{R}}^{d} \) such that \( P \cap A \neq \varnothing \) . Then \( P \cap A \) is also a polytope. | Proof. Any affine subspace \( A \) of \( {\mathbb{R}}^{d} \) is polyhedral. The statement then follows as in the proof of Corollary 9.3. | Yes |
Corollary 9.5. Let \( P \) be a \( d \) -polytope in \( {\mathbb{R}}^{d} \). Then \( P \) has at least \( d + 1 \) facets. | Proof. Let\n\n\[ P = \operatorname{conv}\left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \]\n\nand assume without loss of generality that \( o \in \) int \( P \). Let\n\n\[ Q \mathrel{\text{:=}} \mathop{\bigcap }\limits_{{i = 1}}^{n}K\left( {{x}_{i},1}\right) \]\n\nThen by Theorem 9.1(d), \( P \) and \( Q \) are mutually pola... | Yes |
Corollary 9.6. Let \( P \) be a \( d \) -polytope in \( {\mathbb{R}}^{d} \), let \( {F}_{1},\ldots ,{F}_{n} \) be the facets of \( P \), and let \( K\left( {{x}_{i},{\alpha }_{i}}\right) \) be the supporting halfspace of \( P \) bounded by aff \( {F}_{i} \) for \( i = 1,\ldots, n \) . Then\n\n\[ P = \mathop{\bigcap }\l... | Proof. By Theorem 9.2, \( P \) is polyhedral. Let\n\n\[ P = \mathop{\bigcap }\limits_{{j = 1}}^{m}K\left( {{y}_{j},{\beta }_{j}}\right) \]\n\nbe an irreducible representation of \( P \) . By Theorem 8.2(b),(c), the facets of \( P \) are the sets \( H\left( {{y}_{j},{\beta }_{j}}\right) \cap P \) . But the facets of \( ... | Yes |
Corollary 9.7. Let \( P \) be a d-polytope in \( {\mathbb{R}}^{d} \). Let \( {F}_{j} \) and \( {F}_{k} \) be faces of \( P \) with \[ {F}_{j} \subset {F}_{k} \] and \[ \dim {F}_{j} = j,\;\dim {F}_{k} = k, \] where \[ - 1 \leq j < j + 1 \leq k - 1 < k \leq d. \] Then there are faces \( {F}_{j + 1},\ldots ,{F}_{k - 1} \)... | Proof. With Theorem 9.2 in mind, the statement follows immediately from Corollary 8.6 when \( j \geq 0 \). For \( j = - 1 \), let \( {F}_{0} \) be any vertex of \( {F}_{k} \). If \( k = 1 \), we have the desired conclusion. If \( k \geq 2 \), apply Corollary 8.6 to the faces \( {F}_{0} \) and \( {F}_{k} \). | Yes |
Theorem 9.8. Let \( P \) and \( Q \) be mutually polar d-polytopes in \( {\mathbb{R}}^{d} \) and let \( F \) and \( G \) be conjugate faces of \( P \) and \( Q \), respectively. Then\n\n\[ \n\dim F + \dim G = d - 1.\n\]\n\nIn particular, vertices of \( P \) are conjugate to facets of \( Q \), and facets of \( P \) are ... | Proof. We shall appeal to the proof of Theorem 6.10. As explained there, we need only consider the case where \( F \) and \( G \) are proper faces. Let \( {x}_{1},\ldots ,{x}_{n} \) be the vertices of \( P \), and let \( {x}_{1},\ldots ,{x}_{k} \) be the vertices of \( F \). Then by Theorem 9.1(a),\n\n\[ \nQ = \mathop{... | Yes |
Theorem 10.1. Let \( P \) and \( Q \) be equivalent polytopes with \( \dim P = d \), and let\n\n\[ \varphi : \left( {\mathcal{F}\left( P\right) , \subset }\right) \rightarrow \left( {\mathcal{F}\left( Q\right) , \subset }\right) \]\n\nbe an isomorphism. Then\n\n\[ \dim Q = d \]\n\nand\n\n\[ \dim \varphi \left( F\right)... | Proof. By Corollary 9.7, each face \( F \) of \( P \) is a member of a chain\n\n\[ \varnothing = {F}_{-1} \subsetneq \cdots \subsetneq {F}_{i} \subsetneq \cdots \subsetneq {F}_{d} = P \]\n\n(1)\n\nof faces of \( P \) with\n\n\[ \dim {F}_{i} = i,\;i = - 1,\ldots, d. \]\n\nSince \( \varphi \) is an isomorphism,(1) yields... | Yes |
Theorem 10.2. For any polytope \( P \), there is a dual polytope \( Q \) . | Proof. For any \( d \) -polytope \( P \) there is a \( d \) -polytope \( {P}_{1} \) in \( {\mathbb{R}}^{d} \) with \( o \in \operatorname{int}{P}_{1} \) such that \( P \) and \( {P}_{1} \) are equivalent. Corollary 6.8 shows that \( {Q}_{1} \mathrel{\text{:=}} {P}_{1}^{ \circ } \) is a dual of \( {P}_{1} \) . But then ... | Yes |
Theorem 10.3. Let \( P \) and \( Q \) be dual polytopes with \( \dim P = d \), and let\n\n\[ \psi : \left( {\mathcal{F}\left( P\right) , \subset }\right) \rightarrow \left( {\mathcal{F}\left( Q\right) , \subset }\right) \]\n\nbe an anti-isomorphism. Then\n\n\[ \dim Q = d \]\n\nand\n\n\[ \dim \psi \left( F\right) = d - ... | Proof. As in the proof of Theorem 10.1, we use the fact that each face \( F \) of \( P \) is a member of a chain\n\n\[ \varnothing = {F}_{-1} \subsetneq \cdots \subsetneq {F}_{i} \subsetneq \cdots \subsetneq {F}_{d} = P \]\n\n(4)\n\nof faces of \( P \) with\n\n\[ \dim {F}_{i} = i,\;i = - 1,\ldots, d.\n\n(5)\n\nSince \(... | Yes |
Theorem 10.4. Let \( P \) be a d-polytope, and let \( F \) be a proper face of \( P \). Then \( F \) is the intersection of the facets of \( P \) containing \( F \). If \( F \) is a \( k \)-face, then for \( k = 0,1,\ldots, d - 3 \) there are at least \( d - k \) such facets, for \( k = d - 2 \) there are exactly \( 2\... | Proof. Let \( Q \) be a dual of \( P \), and let \( \psi \) be an anti-isomorphism from \( \left( {\mathcal{F}\left( P\right) , \subset }\right) \) onto \( \left( {\mathcal{F}\left( Q\right) , \subset }\right) \). Let \( F \) be a \( k \)-face of \( P \), and let \( G \mathrel{\text{:=}} \psi \left( F\right) \). Then\n... | Yes |
Theorem 10.5. Let \( P \) be a d-polytope, and let \( x \) be a vertex of \( P \) . Then there are at least \( d \) edges of \( P \) containing \( x \) . | Proof. Let \( Q \) be a dual of \( P \), and let \( \psi \) be an anti-isomorphism from \( \left( {\mathcal{F}\left( P\right) , \subset }\right) \) onto \( \left( {\mathcal{F}\left( Q\right) , \subset }\right) \) . Let \( G \mathrel{\text{:=}} \psi \left( {\{ x\} }\right) \) ; then by Theorem 10.3, \( G \) is a \( \lef... | Yes |
Theorem 11.1. Let \( P \) be a \( d \) -polytope in \( {\mathbb{R}}^{d} \), and let \( H \) be a hyperplane in \( {\mathbb{R}}^{d} \) such that\n\n\[ H \cap \operatorname{int}P \neq \varnothing \text{.}\]\n\nThen the following holds:\n\n(a) The set \( {P}^{\prime } \mathrel{\text{:=}} H \cap P \) is a \( \left( {d - 1}... | Proof. (a) The set \( {P}^{\prime } \) is a polytope by Corollary 9.4. It is clear that the dimension of \( {P}^{\prime } \) is \( d - 1 \).\n\n(b) It follows immediately from the definition of a face that \( {F}^{\prime } \) is a face of \( {P}^{\prime } \), and it is clear that \( \dim {F}^{\prime } \leq \dim F \) . ... | Yes |
Theorem 11.2. Let \( P \) be a \( d \) -polytope in \( {\mathbb{R}}^{d} \), let \( {x}_{0} \) be a vertex of \( P \), and let \( {P}^{\prime } = H \cap P \) be a vertex-figure of \( P \) at \( {x}_{0} \) . Then \( {P}^{\prime } \) is a \( \left( {d - 1}\right) \) -polytope. Furthermore, the mapping \[ F \mapsto {F}^{\p... | Proof. The hyperplane \( H \) intersects \( P \) and is not a supporting hyperplane. Therefore, \[ H \cap \operatorname{int}P \neq \varnothing \] by Theorem 4.1. Then \( {P}^{\prime } \) is a \( \left( {d - 1}\right) \) -polytope by Theorem 11.1(a). It follows from Theorem 11.1(b) that the mapping \[ F \mapsto {F}^{\pr... | Yes |
Corollary 11.3. Let \( P \) be a d-polytope in \( {\mathbb{R}}^{d} \), and let \( {x}_{0} \) be a vertex of \( P \) . Then any two vertex-figures of \( P \) at \( {x}_{0} \) are equivalent. | Proof. In fact, if \( {P}^{\prime } \) and \( {P}^{\prime \prime } \) are vertex-figures of \( P \) at \( {x}_{0} \), then \( \left( {\mathcal{F}\left( {P}^{\prime }\right) , \subset }\right) \) and \( \left( {\mathcal{F}\left( {P}^{\prime \prime }\right) , \subset }\right) \) are both isomorphic to \( \left( {\mathcal... | Yes |
Theorem 11.5. Let \( P \) and \( Q \) be dual d-polytopes, and let\n\n\[ \psi : \left( {\mathcal{F}\left( P\right) , \subset }\right) \rightarrow \left( {\mathcal{F}\left( Q\right) , \subset }\right) \]\n\nbe an anti-isomorphism. Let \( {x}_{0} \) be a vertex of \( P \), and let \( {P}^{\prime } \) be a vertex-figure o... | Proof. We know by Theorem 11.2 that \( \left( {\mathcal{F}\left( {P}^{\prime }\right) , \subset }\right) \) and \( \left( {\mathcal{F}\left( {P/{x}_{0}}\right) , \subset }\right) \) are isomorphic. Taking \( {F}_{1} = \left\{ {x}_{0}\right\} \) and \( {F}_{2} = P \) in the proof of Theorem 11.4, we see that \( \left( {... | Yes |
Theorem 11.6. Let \( P \) be a \( d \) -polytope in \( {\mathbb{R}}^{d} \), let \( {x}_{0} \) be a vertex of \( P \), and let \( {x}_{1},\ldots ,{x}_{k} \) be the vertices of \( P \) adjacent to \( {x}_{0} \) . Let \( H\left( {y,\alpha }\right) \) be a hyperplane in \( {\mathbb{R}}^{d} \) such that \( {x}_{0} \in H\lef... | Proof. Let \( {P}^{\prime } = {H}^{\prime } \cap P \) be a vertex-figure of \( P \) at \( {x}_{0} \), determined by a hyperplane \( {H}^{\prime } \) separating \( {x}_{0} \) from the remaining vertices of \( P \) . Theorem 11.1(b), (d)-or Theorem 11.2-tells that the vertices of \( {P}^{\prime } \) are the 1-point sets ... | Yes |
Corollary 11.7. Let \( P \) be a \( d \) -polytope in \( {\mathbb{R}}^{d} \), let \( {x}_{0} \) be a vertex of \( P \), and let \( {x}_{1},\ldots ,{x}_{k} \) be the vertices of \( P \) adjacent to \( {x}_{0} \) . Then\n\n\[ \operatorname{aff}\left\{ {{x}_{0},{x}_{1},\ldots ,{x}_{k}}\right\} = {\mathbb{R}}^{d}. \] | Proof. If the desired conclusion is not valid, then there is a hyperplane \( H \) containing \( {x}_{0},{x}_{1},\ldots ,{x}_{k} \) . Then both of the two closed halfspaces bounded by \( H \) contain \( {x}_{0},{x}_{1},\ldots ,{x}_{k} \) . By Theorem 11.6 this implies that \( P \) is contained in both of these halfspace... | Yes |
Theorem 11.11. Let \( P \) be a \( d \) -polytope in \( {\mathbb{R}}^{d} \), let \( H \) be a hyperplane in \( {\mathbb{R}}^{d} \) with\n\n\[ H \cap \operatorname{int}P \neq \varnothing ,\;H \cap \operatorname{ext}P = \varnothing ,\]\n\nand let \( K \) be one of the two closed halfspaces bounded by \( H \) . Then we ha... | Proof. (a) This is obvious, cf. Corollary 9.4.\n\n(b) It is obvious that \( {F}^{\prime } \) is a face of \( {P}^{\prime } \) . If \( F \subset K \), then the dimension formula is trivial. If \( F ⊄ K \), then there must be points of \( F \) on both sides of \( H \) ; for if not, then \( H \) would be a supporting hype... | Yes |
Theorem 12.1. Let \( S \) be an e-simplex in \( {\mathbb{R}}^{d} \), and let \( F \) be a proper face of \( S \) . Then \( F \) is also a simplex. | Proof. The vertices of \( F \) are those vertices of \( S \) which are in \( F \) , cf. Theorem 7.3. Any subfamily of an affinely independent family of points is itself affinely independent. Therefore, since \( F \) is the convex hull of its vertices, cf. Theorem 7.2(c), it follows that \( F \) is a simplex. | Yes |
Theorem 12.2. Let \( S \) be an e-simplex in \( {\mathbb{R}}^{d} \), let \( X \) be a non-empty subset of ext \( S \), and let \( F = \operatorname{conv}X \) . Then \( F \) is a face of \( S \), and ext \( F = X \) . | Proof. Let ext \( S = \left\{ {{x}_{1},\ldots ,{x}_{e + 1}}\right\} \), and let us assume that \( X = \left\{ {{x}_{1},\ldots ,{x}_{k}}\right\} \) : To prove that \( F \) is a face of \( S \), we shall show that if \( {y}_{0} \) and \( {y}_{1} \) are two points of \( S \) such that for some \( t \in \rbrack 0,1\lbrack ... | Yes |
Corollary 12.3. Let \( S \) be an e-simplex in \( {\mathbb{R}}^{d} \), and let \( F \) be a \( j \) -face of \( S \), where \( - 1 \leq j \leq e \) . Then for \( k = j,\ldots, e \), the number of \( k \) -faces of \( S \) containing \( F \) equals \[ \left( \begin{array}{l} e - j \\ k - j \end{array}\right) \] | Proof. By Theorems 12.1 and 12.2 there is a one-to-one correspondence between the \( k \) -faces of \( S \) containing \( F \), and the choices of \( \left( {k + 1}\right) - \left( {j + 1}\right) \) vertices from the \( \left( {e + 1}\right) - \left( {j + 1}\right) \) vertices of \( S \) not in \( F \) . This proves th... | Yes |
Corollary 12.4. Let \( S \) be an e-simplex in \( {\mathbb{R}}^{d} \). Then for \( - 1 \leq k \leq e \), the number of \( k \)-faces of \( S \) equals | \[ \left( \begin{array}{l} e + 1 \\ k + 1 \end{array}\right) \] Proof. Take \( j = - 1 \) in Corollary 12.3. | Yes |
Corollary 12.5. Let \( S \) be an e-simplex in \( {\mathbb{R}}^{d} \), and let \( F \) be a \( k \) -face of \( S \), where \( - 1 \leq k \leq e \) . Then the number of facets of \( S \) containing \( F \) equals \( e - k \) . | Proof. It follows from Corollary 12.3 that there are\n\n\[ \left( \begin{matrix} e - k \\ \left( {e - 1}\right) - k \end{matrix}\right) = e - k \]\n\nfacets of \( S \) containing a given \( k \) -face \( F \) . This proves the assertion. | Yes |
Corollary 12.6. Let \( S \) be an e-simplex in \( {\mathbb{R}}^{d} \), and let \( {F}_{1},\ldots ,{F}_{e - k} \) be \( e - k \) facets of \( S \), where \( - 1 \leq k \leq e - 1 \) . Then \( {F}_{1} \cap \cdots \cap {F}_{e - k} \) is a \( k \) -face of \( S \) . | Proof. Let \( {x}_{1},\ldots ,{x}_{e + 1} \) be the vertices of \( S \) . By Theorems 12.1 and 12.2, each \( {F}_{j} \) is the convex hull of certain \( e \) of the \( e + 1 \) vertices. We may assume that\n\n\[ \n{F}_{j} = \operatorname{conv}\left( {\left\{ {{x}_{1},\ldots ,{x}_{e + 1}}\right\} \smallsetminus \left\{ ... | Yes |
Corollary 12.7. Let \( S \) be an e-simplex in \( {\mathbb{R}}^{d} \), and let \( T \) be a dual e-polytope. Then \( T \) is also an e-simplex. | Proof. It follows from Corollary 12.4 that \( S \) has \( e + 1 \) facets. Dually, \( T \) has \( e + 1 \) vertices, cf. Theorem 10.3. But \( e \) -polytopes with \( e + 1 \) vertices are simplices. | Yes |
Corollary 12.8. Let \( P \) be an e-polytope in \( {\mathbb{R}}^{d} \). Then \( P \) is an e-simplex if and only if the number of facets of \( P \) is \( e + 1 \). | Proof. If \( P \) is an \( e \) -simplex, then \( P \) has \( e + 1 \) facets by Corollary 12.4. Conversely, if \( P \) is an \( e \) -polytope with \( e + 1 \) facets, then any dual \( Q \) of \( P \) is an \( e \) -polytope with \( e + 1 \) vertices, cf. Theorem 10.3. Hence, \( Q \) is an \( e \) -simplex, and theref... | Yes |
Theorem 12.9. A d-polytope \( P \) is simplicial if (and only if) each facet of \( P \) is a simplex. | Proof. Let \( F \) be a proper face of \( P \) . By Corollary 9.7 there is a facet \( G \) of \( P \) containing \( F \) . Then \( F \) is a face of \( G \), cf. Theorem 5.2, and since \( G \) is a simplex by assumption, \( F \) is a simplex by Theorem 12.1. | Yes |
Theorem 12.10. Let \( P \) and \( Q \) be dual d-polytopes. Then \( P \) is simple if and only if \( Q \) is simplicial. | Proof. Let \( F \) and \( G \) be proper faces of \( P \) and \( Q \), respectively, corresponding under some anti-isomorphism from \( \left( {\mathcal{F}\left( P\right) , \subset }\right) \) onto \( \left( {\mathcal{F}\left( Q\right) , \subset }\right) \) . Then\n\n\[ \dim G = d - 1 - \dim F \]\n\ncf. Theorem 10.3. Fu... | Yes |
Theorem 12.11. A d-polytope \( P \) is simple if (and only if) each vertex of \( P \) is contained in precisely \( d \) facets. | Proof. Let \( Q \) be a dual of \( P \) . If each vertex of \( P \) is contained in precisely \( d \) facets, then each facet of \( Q \) has precisely \( d \) vertices, cf. Theorem 10.3. Therefore, each facet of \( Q \) is a simplex, whence \( Q \) is simplicial by Theorem 12.9. But then \( P \) is simple by Theorem 12... | Yes |
Theorem 12.12. A d-polytope \( P \) is simple if and only if each vertex of \( P \) is incident to precisely \( d \) edges of \( P \). | Proof. Let \( Q \) be a dual of \( P \), and let \( \psi \) be an anti-isomorphism from \( \left( {\mathcal{F}\left( P\right) , \subset }\right) \) onto \( \left( {\mathcal{F}\left( Q\right) , \subset }\right) \) . Let \( x \) be a vertex of \( P \) . Then the number of edges of \( P \) incident to \( x \) equals the n... | Yes |
Theorem 12.13. A d-polytope \( P \) is simple if and only if each vertex-figure of \( P \) is a simplex. | Proof. Let \( Q \) be a dual of \( P \) . Then the facets of \( Q \) are duals of the vertex-figures of \( P \), cf. Theorem 11.5. The statement then follows from Theorems 12.10 and 12.9. | No |
Theorem 12.14. Let \( P \) be a simple d-polytope, and let \( {F}_{1},\ldots ,{F}_{d - k} \) be \( d - k \) facets of \( P \), where \( 0 \leq k \leq d - 1 \) . Let\n\n\[ F \mathrel{\text{:=}} \mathop{\bigcap }\limits_{{i = 1}}^{{d - k}}{F}_{i} \]\n\nand assume that \( F \neq \varnothing \) . Then \( F \) is a \( k \) ... | Proof. Let \( Q \) be a dual of \( P \), and let \( \psi \) be an anti-isomorphism from \( \left( {\mathcal{F}\left( P\right) , \subset }\right) \) onto \( \left( {\mathcal{F}\left( Q\right) , \subset }\right) \) . By its definition, \( F \) is the largest face contained in the \( {F}_{i} \) ’s, whence \( \psi \left( F... | Yes |
Theorem 12.15. Let \( P \) be a simple d-polytope. Then every proper face of \( P \) is also simple. | Proof. Let \( F \) be a proper face of \( P \), and let \( x \) be a vertex of \( F \) . Letting \( k \mathrel{\text{:=}} \dim F \), we shall prove that there are precisely \( k \) facets of \( F \) containing \( x \) , cf. Theorem 12.11. Let \( Q \) be a dual of \( P \), and let \( \psi \) be an anti-isomorphism from ... | Yes |
Theorem 12.16. Let \( P \) be a simple d-polytope. Then for \( 0 \leq j \leq k \leq d \) there are precisely\n\n\[ \left( \begin{array}{l} d - j \\ d - k \end{array}\right) \]\n\n\( k \) -faces of \( P \) containing a given \( j \) -face of \( P \) . | Proof. For \( k = d \), there is nothing to prove. For \( k < d \), let \( Q \) be a dual of \( P \) , and let \( \psi \) be an anti-isomorphism from \( \left( {\mathcal{F}\left( P\right) , \subset }\right) \) onto \( \left( {\mathcal{F}\left( Q\right) , \subset }\right) \) . Let \( F \) be a given \( j \) -face of \( ... | Yes |
Theorem 12.17. Let \( P \) be a simple d-polytope, let \( {x}_{0} \) be a vertex of \( P \), let \( {x}_{1},\ldots ,{x}_{k} \) be certain \( k \) vertices of \( P \) adjacent to \( {x}_{0} \), and let \( F \) be the smallest face of \( P \) containing \( \left\lbrack {{x}_{0},{x}_{1}}\right\rbrack ,\ldots ,\left\lbrack... | Proof. Let \( Q \) be a dual of \( P \), and let \( \psi \) be an anti-isomorphism from \( \left( {\mathcal{F}\left( P\right) , \subset }\right) \) onto \( \left( {\mathcal{F}\left( Q\right) , \subset }\right) \) . Let \( G \mathrel{\text{:=}} \psi \left( F\right) \) . Then, by duality, \( G \) is the largest face of \... | Yes |
Theorem 12.18. Let \( P \) be a simple d-polytope in \( {\mathbb{R}}^{d} \), and let \( {x}_{0} \) be a vertex of \( P \) . Let \( {P}^{\prime } \) be a d-polytope obtained from \( P \) by truncating the vertex \( {x}_{0} \) . Then \( {P}^{\prime } \) is also a simple d-polytope. Moreover,\n\n\[ \n{f}_{0}\left( {P}^{\p... | Proof. Let \( K \) denote the closed halfspace in \( {\mathbb{R}}^{d} \) such that \( {P}^{\prime } = K \cap P \), and let \( H \) denote the bounding hyperplane of \( K \). \n\nTo see that \( {P}^{\prime } \) is simple, we shall show that each vertex of \( {P}^{\prime } \) is incident to precisely \( d \) edges of \( ... | Yes |
Theorem 12.19. Let \( d \neq 2 \), and let \( P \) be a \( d \) -polytope which is both simple and simplicial. Then \( P \) is a simplex. | Proof. As noted above, we need only consider \( d \geq 3 \) . Let \( {x}_{0} \) be a vertex of \( P \) , and let \( {x}_{1},\ldots ,{x}_{d} \) be the vertices of \( P \) adjacent to \( {x}_{0} \), cf. Theorem 12.12. Let\n\n\[ S \mathrel{\text{:=}} \operatorname{conv}\left\{ {{x}_{0},{x}_{1},\ldots ,{x}_{d}}\right\} \]\... | Yes |
Theorem 13.1. Any hyperplane \( H \) in \( {\mathbb{R}}^{d} \) contains at most \( d \) points from \( {\mathcal{M}}_{d} \) . | Proof. Let \( H = H\left( {y,\alpha }\right) \), where\n\n\[ y = \left( {{\beta }_{1},\ldots ,{\beta }_{d}}\right) \]\n\nThen \( x\left( t\right) \in H\left( {y,\alpha }\right) \) if and only if\n\n\[ {\beta }_{1}t + \cdots + {\beta }_{d}{t}^{d} = \alpha . \]\n\nBy the Fundamental Theorem of Algebra there are at most \... | Yes |
Corollary 13.2. Let \( {t}_{1},\ldots ,{t}_{n} \) be distinct real numbers, where \( n \leq d + 1 \) . Then the n-family \( \left( {x\left( {t}_{1}\right) ,\ldots, x\left( {t}_{n}\right) }\right) \) of points from \( {\mathbb{R}}^{d} \) is affinely independent. | Proof. If \( \left( {x\left( {t}_{1}\right) ,\ldots, x\left( {t}_{n}\right) }\right) \) is affinely dependent, then all the points \( x\left( {t}_{1}\right) ,\ldots, x\left( {t}_{n}\right) \) belong to some affine subspace \( A \) with \( \dim A \leq n - 2 \) . If \( n < d + 1 \), choose \( {t}_{n + 1},\ldots ,{t}_{d +... | Yes |
Theorem 13.3. Let \( P = \operatorname{conv}\left\{ {x\left( {t}_{1}\right) ,\ldots, x\left( {t}_{n}\right) }\right\} \) be a cyclic polytope of type \( C\left( {n, d}\right) \) . Then \( P \) is a \( d \) -polytope. | Proof. By Corollary 13.2, any \( \left( {d + 1}\right) \) -family formed by distinct points \( x\left( {t}_{i}\right) \) is affinely independent. Therefore\n\n\[ \operatorname{aff}\left\{ {x\left( {t}_{1}\right) ,\ldots, x\left( {t}_{d + 1}\right) }\right\} = {\mathbb{R}}^{d}, \]\n\nimplying that \( \dim P = d \) . | Yes |
Theorem 13.4. Let \( P = \operatorname{conv}\left\{ {x\left( {t}_{1}\right) ,\ldots, x\left( {t}_{n}\right) }\right\} \) be a cyclic polytope of type \( C\left( {n, d}\right) \) . Then \[ \text{ext}P = \left\{ {x\left( {t}_{1}\right) ,\ldots, x\left( {t}_{n}\right) }\right\} \text{.} \] | Proof. The inclusion \( \subset \) follows from Theorem 7.2,(a) \( \Rightarrow \) (b). Conversely, to show that \( x\left( {t}_{i}\right) \) is a vertex of \( P \), consider the polynomium \( p\left( t\right) \) of degree 2 defined by \[ p\left( t\right) \mathrel{\text{:=}} - {\left( t - {t}_{i}\right) }^{2} \] \[ = - ... | Yes |
Theorem 13.5. Let \( P = \operatorname{conv}\left\{ {x\left( {t}_{1}\right) ,\ldots, x\left( {t}_{n}\right) }\right\} \) be a cyclic polytope of type \( C\left( {n, d}\right) \). Then \( P \) is simplicial. | Proof. Since \( P \) is a \( d \) -polytope, cf. Theorem 13.3, it suffices to show that any facet of \( P \) is a \( \left( {d - 1}\right) \) -simplex, cf. Theorem 12.9. Let \( F \) be a facet of \( P \). Then the vertices of \( F \) are certain of the vertices of \( P \), say \( x\left( {t}_{{i}_{1}}\right) ,\ldots, x... | Yes |
Theorem 13.7. Let \( P = \operatorname{conv}\left\{ {x\left( {t}_{1}\right) ,\ldots, x\left( {t}_{n}\right) }\right\} \) be a cyclic polytope of type \( C\left( {n, d}\right) \), where \( {t}_{1} < \cdots < {t}_{n} \) . Let \( X \) be a subset of \( \left\{ {x\left( {t}_{1}\right) ,\ldots, x\left( {t}_{n}\right) }\righ... | Proof. The set \( X \) is the set of vertices of a face of \( P \) if and only if there is a facet \( G \) of \( P \) such that\n\n\[X \subset \text{ext}G\text{.}\]\n\n(1)\n\nIn fact, if \( X = \) ext \( F \) for some face \( F \) of \( P \), then by Corollary 9.7 there is a facet \( G \) of \( P \) containing \( F \),... | Yes |
Let \( P = \operatorname{conv}\left\{ {x\left( {t}_{1}\right) ,\ldots, x\left( {t}_{n}\right) }\right\} \) be a cyclic polytope of type \( C\left( {n, d}\right) \), and let \( k \) be an integer such that\n\n\[1 \leq k \leq \lfloor d/2\rfloor \text{.}\]\n\nThen any \( k \) of the points \( x\left( {t}_{1}\right) ,\ldot... | Proof. As in Theorem 13.7, we assume that \( {t}_{1} < \cdots < {t}_{n} \) . When \( k \leq \lfloor d/2\rfloor \) , then \( d - k \geq k \) . Since the number of (odd proper) components of \( X \) cannot exceed the number of points in \( X \), the conclusion follows immediately from Theorem 13.7. | Yes |
Corollary 13.9. Let \( P = \operatorname{conv}\left\{ {x\left( {t}_{1}\right) ,\ldots, x\left( {t}_{n}\right) }\right\} \) and \( Q = \operatorname{conv}\left\{ {x\left( {s}_{1}\right) ,\ldots, x\left( {s}_{n}\right) }\right\} \) be cyclic polytopes, both of type \( C\left( {n, d}\right) \) . Then \( P \) and \( Q \) a... | Proof. We may assume that \( {t}_{1} < \cdots < {t}_{n} \) and \( {s}_{1} < \cdots < {s}_{n} \) . For any face \( F \) of \( P \) with vertices \( x\left( {t}_{{i}_{1}}\right) ,\ldots, x\left( {t}_{{i}_{k}}\right) \), define\n\n\[ \varphi \left( F\right) \mathrel{\text{:=}} \operatorname{conv}\left\{ {x\left( {s}_{{i}_... | Yes |
Theorem 14.2. Let \( P \) be a \( k \) -neighbourly d-polytope, and let \( 1 \leq j < k \) . Then \( P \) is also \( j \) -neighbourly. | Proof. Let \( X \) be any \( j \) -subset of ext \( P \) . Since\n\n\[ \operatorname{card}\left( {\left( {\operatorname{ext}P}\right) \smallsetminus X}\right) \geq d + 1 - j \]\n\n\[ \geq \left( {k - j}\right) + 1 \]\n\nwe see that for any vertex \( x \) of \( P \) not in \( X \) there is a \( k \) -subset \( Y\left( x... | Yes |
Theorem 14.3. Let \( P \) be a \( k \) -neighbourly d-polytope, and let \( X \) be a subset of ext \( P \) containing at least \( k + 1 \) points. Then \( Q \mathrel{\text{:=}} \operatorname{conv}X \) is also \( k \) -neighbourly. | Proof. Let \( Y \) be a \( k \) -subset of ext \( Q = X \) . It follows from the \( k \) -neighbourliness of \( P \) that the set conv \( Y \) is a (proper) face of \( P \) . Being a proper subset of \( Q \) , it must then also be a proper face of \( Q \) . | No |
Theorem 14.4. Let \( P \) be a \( k \) -neighbourly d-polytope. Then every face \( F \) of \( P \) with\n\n\[ 0 \leq \dim F \leq {2k} - 1 \]\n\nis a simplex. | Proof. Let \( j \mathrel{\text{:=}} \dim F \) . Suppose that \( F \) is not a simplex. Then \( F \) has at least \( j + 2 \) vertices. Let \( M \) be a \( \left( {j + 2}\right) \) -subset of ext \( F \) . By Radon’s Theorem, Corollary 2.7, there are non-empty complementary subsets \( {M}_{1} \) and \( {M}_{2} \) of \( ... | Yes |
Corollary 14.5. Let \( P \) be a \( k \) -neighbourly d-polytope, where \( \lfloor d/2\rfloor < k \) . Then \( P \) is a simplex. | Proof. Since \( \lfloor d/2\rfloor < k \) implies \( d \leq {2k} - 1 \), we can apply Theorem 14.4 with \( F = P \) . | No |
Corollary 14.6. Let \( P \) be a \( \left( {d/2}\right) \) -neighbourly \( d \) -polytope, where \( d \) is even. Then \( P \) is simplicial. | Proof. Let \( F \) be a facet of \( P \) . Then \( \dim F = d - 1 = {2k} - 1 \) with \( k = d/2 \) . Theorem 14.4 next shows that \( F \) is a simplex, whence \( P \) is simplicial, cf. Theorem 12.9. | Yes |
Theorem 14.7. A simple d-polytope \( P \) is a dual of a k-neighbourly polytope if and only if any \( k \) facets of \( P \) have a non-empty intersection. | Proof. Let \( Q \) be a dual of \( P \) . Then \( Q \) is a \( d \) -polytope by Theorem 10.3, and \( Q \) is simplicial by Theorem 12.10. By the duality, any \( k \) vertices of \( Q \) belong to a proper face of \( Q \) if and only if any \( k \) facets of \( P \) have a non-empty intersection. But since \( Q \) is s... | No |
Theorem 15.1. For any d-polytope \( P \) in \( {\mathbb{R}}^{d} \), the set of admissible vectors is dense in \( {\mathbb{R}}^{d} \), i.e. for any \( y \in {\mathbb{R}}^{d} \) and any \( \varepsilon > 0 \) there is an admissible vector \( w \) with \( \parallel y - w\parallel < \varepsilon \) . | Proof. We first remark that the union of a finite number of hyperplanes in \( {\mathbb{R}}^{d} \) has no interior points. This follows by repeated application of the observation that for any non-empty open set \( O \) in \( {\mathbb{R}}^{d} \) and any hyperplane \( H \) in \( {\mathbb{R}}^{d} \) , the set \( O \smallse... | Yes |
Theorem 15.2. In a graph \( \mathcal{G}\left( {P, w}\right) \), the top vertex is the only vertex of \( P \) whose in-valance is 0, and the bottom vertex is the only vertex of \( P \) whose out-valence is 0. | Proof. Let \( x \) be a vertex of \( P \) whose in-valence is 0 . Then all the vertices of \( P \) adjacent to \( x \) are below \( x \) . This implies that there is a hyperplane \( H \) with \( w \) as a normal such that \( x \) is above \( H \) and all the vertices adjacent to \( x \) are below \( H \) . Using Theore... | Yes |
Theorem 15.3. Let \( P \) be a \( d \) -polytope in \( {\mathbb{R}}^{d} \), and let \( F \) be a proper face of \( P \) . Then there is an admissible vector \( w \) such that each vertex of \( F \) is above each vertex of \( P \) not in \( F \) . | Proof. Let \( H\left( {y,\alpha }\right) \) be a supporting hyperplane of \( P \) with \( H\left( {y,\alpha }\right) \cap P = F \) , cf. Theorem 7.5. We may assume that \( \langle x, y\rangle \geq \alpha \) for all \( x \in P \) . Let\n\n\[ \gamma \mathrel{\text{:=}} \min \left\{ {\left\langle {{x}^{\prime }, y}\right\... | Yes |
Theorem 15.4. Let \( P \) be a d-polytope in \( {\mathbb{R}}^{d} \), let \( F \) be a proper face of \( P \), and let \( M \) be a subset of ext \( F \) . Then there is an admissible vector \( w \) for \( P \) such that for any vertex \( x \) of \( P \) not in \( M \) there is a path in \( \mathcal{G}\left( P\right) \)... | Proof. By Theorem 15.3 there is an admissible vector \( w \) such that each vertex of \( F \) is above each vertex not in \( F \) . Let \( x \) be any vertex not in \( M \) . When \( x \) is the bottom vertex \( v \), there is nothing to prove. When \( x \neq v \), it follows from Theorem 15.2 that there is at least on... | No |
Theorem 15.5. Let \( P \) be a d-polytope, let \( F \) be a proper face of \( P \), and let \( M \) be a (possibly empty) subset of ext \( F \) . Then the subgraph of \( \mathcal{G}\left( P\right) \) spanned by (ext \( P) \smallsetminus M \) is connected. In particular, \( \mathcal{G}\left( P\right) \) is connected. | Proof. Denoting by \( \Gamma \) the subgraph of \( \mathcal{G}\left( P\right) \) spanned by (ext \( P \) ) \( \smallsetminus M \), Theorem 15.4 shows that there is a vertex \( v \) of \( \Gamma \) such that any vertex of \( \Gamma \) can be joined to \( v \) by a path in \( \Gamma \) . This implies that any two vertice... | Yes |
Theorem 15.7. Let \( \mathcal{S} \) be a connected facet system in a simple d-polytope \( P \) , where \( d \geq 2 \) . Then \( \mathcal{G}\left( \mathcal{S}\right) \) is a \( \left( {d - 1}\right) \) -connected graph. | Proof. We prove the statement by induction on the number \( n \) of members of \( \mathcal{S} \) . For \( n = 1 \), the statement follows immediately from Theorem 15.6. For \( n \geq 2 \), we may number the members \( {F}_{1},\ldots ,{F}_{n} \) of \( \mathcal{S} \) in such a manner that the subsystem \( {\mathcal{S}}^{... | Yes |
Theorem 16.4. Let \( {F}_{1} \) and \( {F}_{2} \) be faces of a polytope \( P \) with \( {F}_{1} \subsetneqq {F}_{2} \) . Then\n\n\[ \mathop{\sum }\limits_{{j \geq - 1}}{\left( -1\right) }^{j}{f}_{j}\left( {{F}_{2}/{F}_{1}}\right) = 0. \] | Proof. We know by Theorem 11.4 that there is a polytope \( Q \) with\n\n\[ \dim Q = \dim {F}_{2} - 1 - \dim {F}_{1} \]\n\nsuch that the lattice \( \left( {\mathcal{F}\left( {{F}_{2}/{F}_{1}}\right) , \subset }\right) \) is isomorphic to the face-lattice \( \left( {\mathcal{F}\left( Q\right) , \subset }\right) \) , and,... | Yes |
Theorem 17.1. For any simple d-polytope \( P \) we have\n\n\[ \mathop{\sum }\limits_{{j = 0}}^{d}{\left( -1\right) }^{j}\left( \begin{array}{l} d - j \\ d - i \end{array}\right) {f}_{j}\left( P\right) = {f}_{i}\left( P\right) \]\n\nfor \( i = 0,\ldots, d \) . | Proof. For any non-empty face \( F \) of \( P \), Euler’s Relation states that\n\n\[ \mathop{\sum }\limits_{{j \geq - 1}}{\left( -1\right) }^{j}{f}_{j}\left( F\right) = 0 \]\n\n(1)\n\nUsing the notation \( {\mathcal{F}}_{i}\left( P\right) \) for the set of \( i \) -faces of \( P \) as we did in the proof of Euler’s Rel... | Yes |
Lemma 17.2. For \( i = 0,\ldots, n \), let \( {P}_{i} \) be a cyclic polytope of type \( C\left( {p + i, d}\right) \) for some fixed \( p \geq d + 1 \) . Then the f-vectors \( f\left( {P}_{i}\right), i = 0,\ldots, n \), form an affinely independent family in \( {\mathbb{R}}^{d} \) . | Proof. The \( f \) -vector of \( {P}_{i} \) has the form\n\n\[ f\left( {P}_{i}\right) = \left( {\left( \begin{matrix} p + i \\ 1 \end{matrix}\right) ,\ldots ,\left( \begin{matrix} p + i \\ n \end{matrix}\right) ,{f}_{n}\left( {P}_{i}\right) ,\ldots ,{f}_{d - 1}\left( {P}_{i}\right) }\right) ,\]\n\ncf. Theorem 13.5 and ... | Yes |
The affine hull aff \( f\left( {\mathcal{P}}_{\sigma }^{d}\right) \) of \( f\left( {\mathcal{P}}_{\sigma }^{d}\right) \) has dimension \( \lfloor d/2\rfloor \) . | Consider the system of \( d + 1 \) (homogeneous) linear equations\n\n\[ \mathop{\sum }\limits_{{j = 0}}^{d}{\left( -1\right) }^{j}\left( \begin{array}{l} d - j \\ d - i \end{array}\right) {x}_{j} = {x}_{i},\;i = 0,\ldots, d, \]\n\n(6)\n\nwith unknowns \( {x}_{0},\ldots ,{x}_{d - 1},{x}_{d} \) . Assigning the value 1 to... | Yes |
Theorem 17.4. The equations\n\n\\[ \n\\mathop{\\sum }\\limits_{{j = 0}}^{d}{\\left( -1\\right) }^{j}\\left( \\begin{array}{l} d - j \\\\ d - i \\end{array}\\right) {x}_{j} = {x}_{i},\\;i = 0,\\ldots, d, \n\\]\n\nwhere \\( {x}_{d} = 1 \\), form a Dehn-Sommerville System. The equations corresponding to odd values of i fo... | In dealing with Dehn-Sommerville Systems it is convenient to use matrix notation. We shall write\n\n\\[ \nx \\mathrel{\\text{:=}} \\left( \\begin{matrix} {x}_{0} \\\\ \\vdots \\\\ {x}_{d - 1} \\\\ {x}_{d} \\end{matrix}\\right) \n\\]\n\nwhere it is always understood that \\( {x}_{d} = 1 \\) . If we let \\( A \\) be the ... | Yes |
Theorem 17.5. The equations\n\n\[ \mathop{\sum }\limits_{{j = 0}}^{d}{\left( -1\right) }^{j}\left( \begin{array}{l} j \\ i \end{array}\right) {x}_{j} = \mathop{\sum }\limits_{{j = 0}}^{d}{\left( -1\right) }^{d + j}\left( \begin{matrix} j \\ d - i \end{matrix}\right) {x}_{j},\;i = 0,\ldots, d, \]\n\nwhere \( {x}_{d} = 1... | Proof. Let \( B \) be the \( \left( {d + 1}\right) \times \left( {d + 1}\right) \) matrix defined by\n\n\[ B \mathrel{\text{:=}} {\left( {\left( -1\right) }^{i + j}\left( \begin{matrix} j \\ i \end{matrix}\right) \right) }_{i = 0,\ldots, d;j = 0,\ldots, d}. \]\n\nNote that \( B \) is invertible, cf. Appendix 3,(11). Si... | Yes |
Theorem 18.1. For any simple d-polytope \( P \) with pfacets we have\n\n\[ \n{f}_{j}\left( P\right) \leq {\Phi }_{j}\left( {d, p}\right) ,\;j = 0,\ldots, d - 2.\n\]\n\nIf \( P \) is dual neighbourly, then\n\n\[ \n{f}_{j}\left( P\right) = {\Phi }_{j}\left( {d, p}\right) ,\;j = 0,\ldots, d - 2.\n\]\n\nIf \( P \) is not d... | Proof. The proof is divided into three parts. In Part A we shall introduce certain numbers \( {g}_{i}\left( P\right) \) associated with a simple \( d \) -polytope \( P \) . In Part \( B \) we shall obtain relations between the numbers \( {g}_{i}\left( P\right) \) and the corresponding numbers \( {g}_{i}\left( F\right) ... | Yes |
Theorem 18.2. (a) The value of \( {\Phi }_{0}\left( {d, p}\right) \) equals\n\n\[ \left( \begin{matrix} p - m - 1 \\ n \end{matrix}\right) + \left( \begin{matrix} p - n - 1 \\ m \end{matrix}\right) \] | Proof. (a) By the definition (2) we have\n\n\[ {\Phi }_{0}\left( {d, p}\right) = \mathop{\sum }\limits_{{i = 0}}^{n}\left( \begin{matrix} p - d + i - 1 \\ i \end{matrix}\right) + \mathop{\sum }\limits_{{i = 0}}^{m}\left( \begin{matrix} p - d + i - 1 \\ i \end{matrix}\right) . \]\n\nThe desired expression then follows e... | No |
Lemma 19.2. Let \( \mathcal{S} \) be a facet system in a simple d-polytope \( P \) such that at least one vertex of \( P \) is not in \( \mathcal{S} \). Then \( \mathcal{S} \) has at least \( d \) external vertices. | Proof. If all vertices of \( \mathcal{S} \) are external, then each member of \( \mathcal{S} \) contributes at least \( d \) external vertices. Suppose that some vertex \( z \) of \( \mathcal{S} \) is internal. By the assumption we also have a vertex \( y \) not in \( \mathcal{S} \). We then use the \( d \) -connectedn... | Yes |
Lemma 19.4. Let \( \mathcal{S} \) be a connected facet system in a simple d-polytope \( P \). Assume that at least one vertex of \( P \) is not in \( \mathcal{S} \), and that \( \mathcal{S} \) has at least two members. Let \( \left( {{x}_{0},{F}_{0}}\right) \) be as in Lemma 19.3. Then at least \( d - 1 \) vertices of ... | Proof. By the connectedness of \( \mathcal{S} \), there is a member \( F \) of \( \mathcal{S} \) with \( F \neq {F}_{0} \) and \( F \cap {F}_{0} \neq \varnothing \). Then by Theorem 12.14, the face \( F \cap {F}_{0} \) has dimension \( d - 2 \), whence \( F \) and \( {F}_{0} \) have at least \( d - 1 \) vertices in com... | Yes |
Lemma 19.5. Let \( \mathcal{S} \) be a facet system in a simple d-polytope \( P \) such that at least one vertex of \( P \) is not in \( \mathcal{S} \) . Then there are at least \( d \) facets \( {G}_{1},\ldots ,{G}_{d} \) of \( P \) such that \( {G}_{1},\ldots ,{G}_{d} \) are not in \( \mathcal{S} \) but each contains... | Proof. Let \( x \) be a vertex of \( P \) not in \( \mathcal{S} \) . Let \( Q \) be a dual of \( P \) in \( {\mathbb{R}}^{d} \), and let \( \psi \) be an anti-isomorphism from \( \left( {\mathcal{F}\left( P\right) , \subset }\right) \) onto \( \left( {\mathcal{F}\left( Q\right) , \subset }\right) \) . Writing\n\n\[ \ma... | No |
For any simplicial d-polytope \( P \) with \( p \) vertices we have\n\n\[ \n{f}_{j}\left( P\right) \geq {\varphi }_{d - 1 - j}\left( {d, p}\right) ,\;j = 1,\ldots, d - 1.\n\]\n\nMoreover, there are simplicial d-polytopes \( P \) with \( p \) vertices such that\n\n\[ \n{f}_{j}\left( P\right) = {\varphi }_{d - 1 - j}\lef... | Equality in Corollary 19.7 is attained by the duals of the truncation polytopes, and, for \( d \geq 4 \), only by these. They are the polytopes obtained from simplices by successive addition of pyramids over facets; they are called stacked polytopes. | Yes |
Proposition 1.4.1. Let \( X \) be a topological space and suppose that \( f \) maps \( X \) onto \( Y \) . Let \( \mathcal{T} \) be any topology on \( Y \) and let \( {\mathcal{T}}_{f} \) be the quotient topology on \( Y \) induced by \( f \) .\n\n(1) If \( f : X \rightarrow \left( {Y,\mathcal{T}}\right) \) is continuo... | Proof. Suppose that \( f : X \rightarrow \left( {Y,\mathcal{T}}\right) \) is continuous. If \( V \) is in \( \mathcal{T} \), then \( {f}^{-1}\left( V\right) \) is in open in \( X \) and so \( V \) is in \( {\mathcal{T}}_{f} \) . If, in addition, \( f : X \rightarrow \left( {Y,\mathcal{T}}\right) \) is an open map then ... | Yes |
Proposition 1.4.2. Suppose that \( f \) maps \( X \) into \( Y \) where \( X \) and \( Y \) are topological spaces, \( Y \) having the quotient topology \( {\mathcal{T}}_{f} \) . For each map \( g : Y \rightarrow Z \) define \( {g}_{1} : X \rightarrow Z \) by \( {g}_{1} = {gf} \) . Then \( g \) is continuous if and onl... | Proof. As \( f \) is continuous, the continuity of \( g \) implies that of \( {g}_{1} \) . Now suppose that \( {g}_{1} \) is continuous. For an open subset \( V \) of \( Z \) (we assume, of course, that \( Z \) is a topological space) we have\n\n\[{\left( {g}_{1}\right) }^{-1}\left( V\right) = {f}^{-1}\left( {{g}^{-1}V... | Yes |
Theorem 1.5.3. If \( H \) is a normal subgroup of a topological group \( G \), then \( G/H \) with the usual structures is a topological group. | For a proof and for further information, see [20], [23], [39], [67], [69] and [94]. | No |
Theorem 2.5.1. Let \( A \) be in \( \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \) . The following statements are equivalent and characterize elements of \( \mathrm{{SU}}\left( {2,\mathbb{C}}\right) \) .\n\n(i) \( A \) is unitary;\n\n(ii) \( \parallel A{\parallel }^{2} = 2 \) ;\n\n(iii) \( A \) is a quaternion.\n\nIn par... | Proof. Suppose that\n\n\[ A = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) ,\;{ad} - {bc} = 1, \]\n\nthen\n\n\[ A{A}^{ * } = \left( \begin{matrix} {\left| a\right| }^{2} + {\left| b\right| }^{2} & a\bar{c} + b\bar{d} \\ \bar{a}c + \bar{b}d & {\left| c\right| }^{2} + {\left| d\right| }^{2} \end{matrix}\rig... | Yes |
Theorem 3.1.4. A function \( \phi \) is a Euclidean isometry if and only if it is of the form\n\n\[ \phi \left( x\right) = {xA} + {x}_{0} \]\n\nwhere \( A \) is an orthogonal matrix and \( {x}_{0} \in {\mathbb{R}}^{n} \) . | Proof. As an orthogonal matrix preserves lengths, it is clear that any \( \phi \) of the given form is an isometry. Conversely, if \( \phi \) is an isometry, then \( \phi \left( x\right) - \phi \left( 0\right) \) is an isometry which fixes the origin and so is given by an orthogonal matrix (as in the proof of Theorem 3... | Yes |
Theorem 3.2.1. Let \( \phi \) be any Möbius transformation and \( \sum \) any sphere. Then \( \phi \left( \sum \right) \) is also a sphere. | Proof. It is easy to see that \( \phi \left( \sum \right) \) is a sphere whenever \( \phi \) is a Euclidean isometry: in particular, this holds when \( \phi \) is the reflection in a plane. It is equally easy to see that \( \phi \left( \sum \right) \) is a sphere when \( \phi \left( x\right) = {kx}, k > 0 \) .\n\nEach ... | Yes |
Theorem 3.2.3. For any Möbius transformation \( \phi \) and any spheres \( \sum \) and \( {\sum }^{\prime } \), \[ \left( {\phi \left( \sum \right) ,\phi \left( {\sum }^{\prime }\right) }\right) = \left( {\sum ,{\sum }^{\prime }}\right) . \] | Proof. A Möbius transformation maps a sphere \( \sum \) to a sphere \( {\sum }^{\prime } \) and so induces a map \[ \left( {{a}_{0},{a}_{1},\ldots ,{a}_{n},{a}_{n + 1}}\right) \mapsto \left( {{a}_{0}^{\prime },{a}_{1}^{\prime },\ldots ,{a}_{n}^{\prime },{a}_{n + 1}^{\prime }}\right) \] between the coefficient vectors (... | Yes |
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