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1,900 | Piercing convex sets | math.MG | A family of sets has the $(p,q)$ property if among any $p$ members of the
family some $q$ have a nonempty intersection. It is shown that for every $p\ge
q\ge d+1$ there is a $c=c(p,q,d)<\infty$ such that for every family $\scr F$ of
compact, convex sets in $R^d$ that has the $(p,q)$ property there is a set of
at most $... | math |
1,901 | A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere | math.MG | We describe a characterization of convex polyhedra in $\h^3$ in terms of
their dihedral angles, developed by Rivin. We also describe some geometric and
combinatorial consequences of that theory. One of these consequences is a
combinatorial characterization of convex polyhedra in $\E^3$ all of whose
vertices lie on the ... | math |
1,902 | On the Busemann-Petty problem concerning central sections of centrally symmetric convex bodies | math.MG | We present a method which shows that in $\Eb$ the Busemann-Petty problem,
concerning central sections of centrally symmetric convex bodies, has a
positive answer. Together with other results, this settles the problem in each
dimension. | math |
1,903 | Average kissing numbers for non-congruent sphere packings | math.MG | The Koebe circle packing theorem states that every finite planar graph can be
realized as the nerve of a packing of (non-congruent) circles in R^3. We
investigate the average kissing number of finite packings of non-congruent
spheres in R^3 as a first restriction on the possible nerves of such packings.
We show that th... | math |
1,904 | The Construction of Self-Similar Tilings | math.MG | We give a construction of a self-similar tiling of the plane with any
prescribed expansion coefficient $\lambda\in\C$ (satisfying the necessary
algebraic condition of being a complex Perron number).
For any integer $m>1$ we show that there exists a self-similar tiling with
$2\pi/m$-rotational symmetry group and expan... | math |
1,905 | The Number of Intersection Points Made by the Diagonals of a Regular Polygon | math.MG | We give a formula for the number of interior intersection points made by the
diagonals of a regular $n$-gon. The answer is a polynomial on each residue
class modulo 2520. We also compute the number of regions formed by the
diagonals, by using Euler's formula $V-E+F=2$. | math |
1,906 | Realization spaces of 4-polytopes are universal | math.MG | Let $P\subset\R^d$ be a $d$-dimensional polytope. The {\em realization space}
of~$P$ is the space of all polytopes $P'\subset\R^d$ that are combinatorially
equivalent to~$P$, modulo affine transformations. We report on work by the
first author, which shows that realization spaces of \mbox{4-dimensional}
polytopes can b... | math |
1,907 | Highly saturated packings and reduced coverings | math.MG | We introduce and study certain notions which might serve as substitutes for
maximum density packings and minimum density coverings. A body is a compact
connected set which is the closure of its interior. A packing $\cal P$ with
congruent replicas of a body $K$ is $n$-saturated if no $n-1$ members of it can
be replaced ... | math |
1,908 | Uniformly distributed distances: A geometric application of Jansen's inequality | math.MG | Let $d_1\leq d_2\leq\ldots\leq d_{n\choose 2}$ denote the distances
determined by $n$ points in the plane. It is shown that $\min\sum_i
(d_{i+1}-d_i)^2=O(n^{-6/7})$, where the minimum is taken over all point sets
with minimal distance $d_1 \geq 1$. This bound is asymptotically tight. | math |
1,909 | Metric Entropy of Homogeneous Spaces | math.MG | For a (compact) subset $K$ of a metric space and $\varepsilon > 0$, the {\em
covering number} $N(K , \varepsilon )$ is defined as the smallest number of
balls of radius $\varepsilon$ whose union covers $K$. Knowledge of the {\em
metric entropy}, i.e., the asymptotic behaviour of covering numbers for
(families of) metri... | math |
1,910 | Another low-technology estimate in convex geometry | math.MG | We give a short argument that for some C > 0, every n-dimensional Banach ball
K admits a 256-round subquotient of dimension at least C n/(log n). This is a
weak version of Milman's quotient of subspace theorem, which lacks the
logarithmic factor. | math |
1,911 | Almost-tiling the plane by ellipses | math.MG | For any delta > 1 we construct a periodic and locally finite packing of the
plane with ellipses whose delta-enlargement covers the whole plane. This
answers a question of Imre B\'ar\'any. On the other hand, we show that if C is
a packing in the plane with circular discs of radius at most 1, then its
1.00001-enlargement... | math |
1,912 | Extremal Approximately Convex Functions and Estimating the Size of Convex Hulls | math.MG | A real valued function $f$ defined on a convex $K$ is anemconvex function iff
it satisfies $$ f((x+y)/2) \le (f(x)+f(y))/2 + 1. $$ A thorough study of
approximately convex functions is made. The principal results are a sharp
universal upper bound for lower semi-continuous approximately convex functions
that vanish on t... | math |
1,913 | Circumscribing constant-width bodies with polytopes | math.MG | Makeev conjectured that every constant-width body is inscribed in the dual
difference body of a regular simplex. We prove that homologically, there are an
odd number of such circumscribing bodies in dimension 3, and therefore
geometrically there is at least one. We show that the homological answer is
zero in higher dim... | math |
1,914 | Rotations of the three-sphere and symmetry of the Clifford torus | math.MG | We describe decomposition formulas for rotations of $R^3$ and $R^4$ that have
special properties with respect to stereographic projection. We use the lower
dimensional decomposition to analyze stereographic projections of great circles
in $S^2 \subset R^3$. This analysis provides a pattern for our analysis of
stereogra... | math |
1,915 | An overview of the Kepler conjecture | math.MG | This is the first in a series of papers giving a proof of the Kepler
conjecture, which asserts that the density of a packing of congruent spheres in
three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This
is the oldest problem in discrete geometry and is an important part of
Hilbert's 18th proble... | math |
1,916 | A formulation of the Kepler conjecture | math.MG | This is the second in a series of papers giving a proof of the Kepler
conjecture, which asserts that the density of a packing of congruent spheres in
three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This
is the oldest problem in discrete geometry and is an important part of
Hilbert's 18th probl... | math |
1,917 | Sphere packings I | math.MG | We describe a program to prove the Kepler conjecture on sphere packings. We
then carry out the first step of this program. Each packing determines a
decomposition of space into Delaunay simplices, which are grouped together into
finite configurations called Delaunay stars. A score, which is related to the
density of pa... | math |
1,918 | Sphere packings II | math.MG | An earlier paper describes a program to prove the Kepler conjecture on sphere
packings. This paper carries out the second step of that program. A sphere
packing leads to a decomposition of $R^3$ into polyhedra. The polyhedra are
divided into two classes. The first class of polyhedra, called quasi-regular
tetrahedra, ha... | math |
1,919 | Sphere packings III | math.MG | This is the fifth in a series of papers giving a proof of the Kepler
conjecture, which asserts that the density of a packing of congruent spheres in
three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This
is the oldest problem in discrete geometry and is an important part of
Hilbert's 18th proble... | math |
1,920 | Sphere packings IV | math.MG | This is the sixth in a series of papers giving a proof of the Kepler
conjecture, which asserts that the density of a packing of congruent spheres in
three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This
is the oldest problem in discrete geometry and is an important part of
Hilbert's 18th proble... | math |
1,921 | Sphere packings V | math.MG | The Hales program to prove the Kepler conjecture on sphere packings consists
of five steps, which if completed, will jointly comprise a proof of the
conjecture. We carry out step five of the program [outlined in
math.MG/9811073], a proof that the local density of a certain combinatorial
arrangement, the pentahedral pri... | math |
1,922 | The Kepler conjecture | math.MG | This is the eighth and final paper in a series giving a proof of the Kepler
conjecture, which asserts that the density of a packing of congruent spheres in
three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This
is the oldest problem in discrete geometry and is an important part of
Hilbert's 18th... | math |
1,923 | A proof of the dodecahedral conjecture | math.MG | The dodecahedral conjecture states that the volume of the Voronoi polyhedron
of a sphere in a packing of equal spheres is at least the volume of a regular
dodecahedron with inradius 1. The authors prove the conjecture following the
methodology of the proof the Kepler conjecture. (See math.MG/9811071.) | math |
1,924 | Möbius invariants for pairs of spheres | math.MG | In this article we construct a complete system of M\"obius-geometric
invariants for pairs $(S^m, S^l), l \leq m$, of spheres contained in the
M\"obius space $S^n$. It consists of n-m generalised stationary angles. We
interpret these invariants geometrically. | math |
1,925 | On traces of $d$-stresses in the skeletons of lower dimensions of homology $d$-manifolds | math.MG | We show how a $d$-stress on a piecewise-linear realization of an oriented
(non-simplicial, in general) $d$-manifold in \rd naturally induces stresses of
lower dimensions on this manifold, and discuss implications of this
construction to the analysis of self-stresses in spatial frameworks. The
constructed mappings are n... | math |
1,926 | An analytic solution to the Busemann-Petty problem on sections of convex bodies | math.MG | We derive a formula connecting the derivatives of parallel section functions
of an origin-symmetric star body in R^n with the Fourier transform of powers of
the radial function of the body. A parallel section function (or
(n-1)-dimensional X-ray) gives the ((n-1)-dimensional) volumes of all
hyperplane sections of the b... | math |
1,927 | A positive solution to the Busemann-Petty problem in R^4 | math.MG | H. Busemann and C. M. Petty posed the following problem in 1956: If K and L
are origin-symmetric convex bodies in R^n and for each hyperplane H through the
origin the volumes of their central slices satisfy vol(K cap H) < vol(L cap H),
does it follow that the volumes of the bodies themselves satisfy vol(K) <
vol(L)?
... | math |
1,928 | Discrete versions of the Beckman-Quarles theorem | math.MG | TO APPEAR IN AEQUATIONES MATHEMATICAE - WITHOUT THEOREM 2. THEOREM 2 IS
CORRECTLY PROVED IN PREVIOUS VERSIONS 1 AND 2. AUTHOR'S VERSION 3 (WITH A NEW
FIGURE 6A) IS UNNECESSARY. Let F \subseteq R denote the field of numbers which
are constructible by means of ruler and compass. We prove that: (1) if x,y \in
R^n (n>1) an... | math |
1,929 | Classification of Finite Subspaces of Metric Space Instead of Constraints on Metrics | math.MG | A new method of metric space investigation, based on classification of its
finite subspaces, is suggested. It admits to derive information on metric space
properties which is encoded in metric. The method describes geometry in terms
of only metric. It admits to remove constraints imposed usually on metric (the
triangle... | math |
1,930 | Continuous rotation invariant valuations on convex sets | math.MG | The famous Hadwiger theorem classifies all rigid motion invariant continuous
valuations on convex sets as linear conbinations of quermassintegrals. We prove
much more general result. We classify continuous valuations which are invariant
with respect to the orthogonal (or special orthogonal) group. Some applications
to ... | math |
1,931 | Menger curvature and rectifiability | math.MG | For a Borel set E in R^n, the total Menger curvature of E, or c(E), is the
integral over E^3 (with respect to 1-dimensional Hausdorff measure in each
factor of E) of c(x,y,z)^2, where 1/c(x,y,z) is the radius of the circle
passing through three points x, y, and z in E.
Let H^1(X) denote the 1-dimensional Hausdorff me... | math |
1,932 | A discrete form of the Beckman-Quarles theorem for rational eight-space | math.MG | Let Q denote the field of rational numbers. Let F \subseteq R is a euclidean
field. We prove that: (1) if x,y \in F^n (n>1) and |x-y| is constructible by
means of ruler and compass then there exists a finite set S(x,y) \subseteq F^n
containing x and y such that each map from S(x,y) to R^n preserving unit
distance prese... | math |
1,933 | The Honeycomb Conjecture | math.MG | The classical honeycomb conjecture asserts that any partition of the plane
into regions of equal area has perimeter at least that of the regular hexagonal
honeycomb tiling. Pappus discusses this problem in his preface to Book V. This
paper gives the first general proof of the conjecture.
The revision is the published... | math |
1,934 | Higher dimensional flexible polyhedra | math.MG | In the previous version of the paper it was announced that ``sphere
homeomorphic flexible polyhedra (with self intersections) do really exist in
n-dimensional Euclidean, Lobachevskij and spherical spaces for each $n\geq
3$.'' Now the paper has been withdrawn by the author, due a crucial error in
the proof of the main t... | math |
1,935 | Notions of denseness | math.MG | The notion of a completely saturated packing [Fejes Toth, Kuperberg and
Kuperberg, Highly saturated packings and reduced coverings, Monats. Math. 125
(1998) 127-145] is a sharper version of maximum density, and the analogous
notion of a completely reduced covering is a sharper version of minimum
density. We define two ... | math |
1,936 | Tangent Spheres and Triangle Centers | math.MG | Any four mutually tangent spheres in R^3 determine three coincident lines
through opposite pairs of tangencies. As a consequence, we define two new
triangle centers. | math |
1,937 | Densest Lattice Packings of 3-Polytopes | math.MG | Based on Minkowski's work on critical lattices of 3-dimensional convex bodies
we present an efficient algorithm for computing the density of a densest
lattice packing of an arbitrary 3-polytope. As an application we calculate
densest lattice packings of all regular and Archimedean polytopes. | math |
1,938 | On arithmetic Kleinian groups generated by three half-turns | math.MG | We study a generalization of the Fuchsian triangle groups to the hyperbolic
3-space, namely, the groups generated by half-turns in three hyperbolic lines.
The role of the hyperbolic triangles is now played by the right-angled
hexagons. This class of groups is very close to the arbitrary 2-generator
Kleinian groups but ... | math |
1,939 | Open sets satisfying systems of congruences | math.MG | A famous result of Hausdorff states that a sphere with countably many points
removed can be partitioned into three pieces A,B,C such that A is congruent to
B (i.e., there is an isometry of the sphere which sends A to B), B is congruent
to C, and A is congruent to (B union C); this result was the precursor of the
Banach... | math |
1,940 | Drawing with Complex Numbers | math.MG | It is not commonly realized that the algebra of complex numbers can be used
in an elegant way to represent the images of ordinary 3-dimensional figures,
orthographically projected to the plane. We describe these ideas here, both
using simple geometry and setting them in a broader context. | math |
1,941 | Location of incenters and Fermat points in variable triangles | math.MG | The orthocentroidal circle of a nonequilateral triangle has diameter GH,
joining the centroid to the orthocenter. We show that the incenters of
triangles with a given Euler line simply cover the interior of the
orthocentroidal circle, and that their Fermat points also lie within this
circle. | math |
1,942 | Lattice Substitution Systems and Model Sets | math.MG | The paper studies ways in which the sets of a partition of a lattice in
$\RR^n$ become regular model sets. The main theorem gives equivalent conditions
which assure that a matrix substitution system on a lattice in $\RR^n$ gives
rise to regular model sets (based on $p$-adic-like internal spaces), and hence
to pure poin... | math |
1,943 | Model sets: a survey | math.MG | This article surveys the mathematics of the cut and project method as applied
to point sets, called here {\em model sets}. It covers the geometric,
arithmetic, and analytical sides of this theory as well as diffraction and the
connection with dynamical systems. | math |
1,944 | Bounds for Local Density of Sphere Packings and the Kepler Conjecture | math.MG | This paper describes the local density inequality approach to getting upper
bounds for sphere packing densities in R^n. This approach was first suggested
by L. Fejes-Toth in 1956 to prove the Kepler conjecture that the densest sphere
packing in R^3 is the "cannonball packing". The approaches of L. Fejes-Toth,
W.-Y. Hsi... | math |
1,945 | A direct proof of a theorem of Blaschke and Lebesgue | math.MG | The Blaschke-Lebesgue Theorem states that among all planar convex domains of
given constant width B the Reuleaux triangle has minimal area. It is the
purpose of the present note to give a direct proof of this theorem by analyzing
the underlying variational problem. The advantages of the proof are that it
shows uniquene... | math |
1,946 | Real Analysis, Quantitative Topology, and Geometric Complexity | math.MG | Contents
1 Mappings and distortion
2 The mathematics of good behavior much of the time, and the BMO frame of
mind
3 Finite polyhedra and combinatorial parameterization problems
4 Quantitative topology, and calculus on singular spaces
5 Uniform rectifiability
Appendices
A Fourier transform calculations
B... | math |
1,947 | Helly-type Theorems for Plane Convex Curves | math.MG | Families of translates and homothets of strictly convex curves are proven to
possess Helly-type properties generalizing those of a circle. Weaker results
are shown for arbitrary convex curves. | math |
1,948 | Trilinear relations between the tetrahedron volumes in a cluster | math.MG | We prove elegant trilinear formulas connecting products of volumes of
Euclidean tetrahedra with vertices taken from a given set of 6 points. We
propose a way for generalizing those formulas. | math |
1,949 | Beyond the Descartes circle theorem | math.MG | The Descartes circle theorem states that if four circles are mutually tangent
with disjoint intersion, then their curvatures (or "bends) b_j = 1/r_j satisfy
the relation (b_1 + b_2 + b_3 + b_4)^2 = 2(b_1^2 + b_2^2 + b_3^2 + b_4^2). We
show that similar relations hold involving the centers of the circles in such a
confi... | math |
1,950 | Discrete versions of the Beckman-Quarles theorem from the definability results of Raphael M. Robinson | math.MG | We derive author's discrete forms of the Beckman-Quarles theorem from the
definability results of Raphael M. Robinson. | math |
1,951 | Minkowski- versus Euclidean rank for products of metric spaces | math.MG | We introduce a notion of the Euclidean- and the Minkowski rank for arbitrary
metric spaces and we study their behaviour with respect to products. We show
that the Minkowski rank is additive with respect to metric products, while
additivity of the Euclidean rank only holds under additional assumptions, e.g.
for Riemanni... | math |
1,952 | Geometry without topology as a new conception of geometry | math.MG | A geometric conception is a method of a geometry construction. The Riemannian
geometric conception and a new T-geometric one are considered. T-geometry is
built only on the basis of information included in the metric (distance between
two points). Such geometric concepts as dimension, manifold, metric tensor,
curve are... | math |
1,953 | Local Complexity of Delone Sets and Crystallinity | math.MG | This paper characterizes when a Delone set X is an ideal crystal in terms of
restrictions on the number of its local patches of a given size or on the
hetereogeneity of their distribution. Let N(T) count the number of
translation-inequivalent patches of radius T in X and let M(T) be the minimum
radius such that every c... | math |
1,954 | Pushing disks apart - The Kneser-Poulsen conjecture in the plane | math.MG | We give a proof of the planar case of a longstanding conjecture of Kneser
(1955) and Poulsen (1954). In fact, we prove more by showing that if a finite
set of disks in the plane is rearranged so that the distance between each pair
of centers does not decrease, then the area of the union does not decrease, and
the area ... | math |
1,955 | The hypermetric cone on seven vertices | math.MG | The hypermetric cone $HYP_n$ is the set of vectors $(d_{ij})_{1\leq i< j\leq
n}$ satisfying the inequalities $\sum_{1\leq i<j\leq n} b_ib_jd_{ij}\leq 0 with
b_i\in\Z and \sum_{i=1}^{n}b_i=1$. A Delaunay polytope of a lattice is called
extremal if the only affine bijective transformations of it into a Delaunay
polytope,... | math |
1,956 | A proof of Atiyah's conjecture on configurations, of four points in Euclidean three-space | math.MG | From any configuration of finitely many points in Euclidean three-space,
Atiyah constructed a determinant and conjectured that it was always non-zero.
Atiyah and Sutcliffe (hep-th/0105179) amass a great deal of evidence it its
favour. In this article we prove the conjecture for the case of four points. | math |
1,957 | New upper bounds on sphere packings I | math.MG | We develop an analogue for sphere packing of the linear programming bounds
for error-correcting codes, and use it to prove upper bounds for the density of
sphere packings, which are the best bounds known at least for dimensions 4
through 36. We conjecture that our approach can be used to solve the sphere
packing proble... | math |
1,958 | New upper bounds on sphere packings II | math.MG | We continue the study of the linear programming bounds for sphere packing
introduced by Cohn and Elkies. We use theta series to give another proof of the
principal theorem, and present some related results and conjectures. | math |
1,959 | Partially Paradoxist Smarandache Geometries | math.MG | A paradoxist Smarandache geometry combines Euclidean, hyperbolic, and
elliptic geometry into one space along with other non-Euclidean behaviors of
lines that would seem to require a discrete space. A class of continuous spaces
is presented here together with specific exmples that exhibit almost all of
these phenomena a... | math |
1,960 | Improving Rogers' upper bound for the density of unit ball packings via estimating the surface area of Voronoi cells from below in Euclidean d-space for all d>7 | math.MG | The sphere packing problem asks for the densest packing of unit balls in
d-dimensional Euclidean space. This problem has its roots in geometry, number
theory and it is part of Hilbert's 18th problem. In 1958 C. A. Rogers proved a
non-trivial upper bound for the density of unit ball packings in d-dimensional
Euclidean s... | math |
1,961 | On the existence of completely saturated packings and completely reduced covering | math.MG | A packing by a body $K$ is collection of congruent copies of $K$ (in either
Euclidean or hyperbolic space) so that no two copies intersect nontrivially in
their interiors. A covering by $K$ is a collection of congruent copies of $K$
such that for every point $p$ in the space there is copy in the collection
containing $... | math |
1,962 | An extremum property characterizing the n-dimensional regular cross-polytope | math.MG | In the spirit of the Genetics of the Regular Figures, by L. Fejes T\'oth, we
prove the following theorem: If $2n$ points are selected in the $n$-dimensional
Euclidean ball $B^n$ so that the smallest distance between any two of them is
as large as possible, then the points are the vertices of an inscribed regular
cross-... | math |
1,963 | A Note on Shelling | math.MG | The radial distribution function is a characteristic geometric quantity of a
point set in Euclidean space that reflects itself in the corresponding
diffraction spectrum and related objects of physical interest. The underlying
combinatorial and algebraic structure is well understood for crystals, but less
so for non-per... | math |
1,964 | Successive Minima and Lattice Points | math.MG | The main purpose of this note is to prove an upper
bound on the number of lattice points of a centrally symmetric
convex body in terms of the successive minima of the body. This bound
improves on former bounds and narrows the gap towards a lattice
point analogue of Minkowski's second theorem on successive minim... | math |
1,965 | The Beckman-Quarles theorem for continuous mappings from R^n to C^n | math.MG | Let \phi((x_1,...,x_n),(y_1,...,y_n))=(x_1-y_1)^2+...+(x_n-y_n)^2. We say
that f:R^n -> C^n preserves distance d>=0 if for each x,y \in R^n \phi(x,y)=d^2
implies \phi(f(x),f(y))=d^2. We prove that if x,y \in R^n (n>=3) and
|x-y|=(\sqrt{2+2/n})^k \cdot (2/n)^l (k,l are non-negative integers) then there
exists a finite s... | math |
1,966 | Cubic Polyhedra | math.MG | A cubic polyhedron is a polyhedral surface whose edges are exactly all the
edges of the cubic lattice. Every such polyhedron is a discrete minimal
surface, and it appears that many (but not all) of them can be relaxed to
smooth minimal surfaces (under an appropriate smoothing flow, keeping their
symmetries). Here we gi... | math |
1,967 | Sphere Packings in 3 Dimensions | math.MG | This short note describes the tentative form of a finite-dimensional
optimization problem that may be of use in a second-generation proof of the
Kepler conjecture. In the original 1998 proof of the Kepler conjecture, the
form of the optimization problem was constrained by limits to computer power
and by the speed of th... | math |
1,968 | Kolakoski-(3,1) is a (deformed) model set | math.MG | Unlike the (classical) Kolakoski sequence on the alphabet {1,2}, its analogue
on {1,3} can be related to a primitive substitution rule. Using this
connection, we prove that the corresponding bi-infinite fixed point is a
regular generic model set and thus has a pure point diffraction spectrum. The
Kolakoski-(3,1) sequen... | math |
1,969 | The Beckman-Quarles theorem for continuous mappings from R^2 to C^2 | math.MG | Let \phi((x_1,x_2),(y_1,y_2))=(x_1-y_1)^2+(x_2-y_2)^2. We say that f:R^2 ->
C^2 preserves distance d>=0 if for each x,y \in R^2 \phi(x,y)=d^2 implies
\phi(f(x),f(y))=d^2. We prove that if x,y \in R^2 and |x-y|=(2\sqrt{2}/3)^k
\cdot (\sqrt{3})^l (k,l are non-negative integers) then there exists a finite
set {x,y} \subse... | math |
1,970 | Products of hyperbolic metric spaces | math.MG | Let (X_i,d_i), i=1,2, be proper geodesic hyperbolic metric spaces. We give a
general construction for a ``hyperbolic product'' X_1{times}_h X_2 which is
itself a proper geodesic hyperbolic metric space and examine its boundary at
infinity. | math |
1,971 | Non Standard Metric Products | math.MG | We consider a fairly general class of natural non standard metric products
and classify those amongst them, which yield a product of certain type (for
instance an inner metric space) for all possible choices of factors of this
type (inner metric spaces). We further prove the additivity of the Minkowski
rank for a large... | math |
1,972 | The Honeycomb Problem on the Sphere | math.MG | The honeycomb problem on the sphere asks for the perimeter-minimizing
partition of the sphere into N equal areas. This article solves the problem
when N=12. The unique minimizer is a tiling of 12 regular pentagons in the
dodecahedral arrangement. | math |
1,973 | Optimally dense packings of hyperbolic space | math.MG | In previous work a probabilistic approach to controlling difficulties of
density in hyperbolic space led to a workable notion of optimal density for
packings of bodies. In this paper we extend an ergodic theorem of Nevo to
provide an appropriate definition of optimal dense packings. Examples are given
to illustrate var... | math |
1,974 | Products of hyperbolic metric spaces II | math.MG | In arXiv math.MG/0207296 we introduced a product construction for locally
compact, complete, geodesic hyperbolic metric spaces. In the present paper we
define the hyperbolic product for general Gromov-hyperbolic spaces. In the case
of roughly geodesic spaces we also analyse the boundary at infinity. | math |
1,975 | On the volume of spherical Lambert cube | math.MG | The calculation of volumes of polyhedra in the three-dimensional Euclidean,
spherical and hyperbolic spaces is very old and difficult problem. In
particular, an elementary formula for volume of non-euclidean simplex is still
unknown. One of the simplest polyhedra is the Lambert cube
Q(\alpha,\beta,\gamma). By definitio... | math |
1,976 | Intrinsic L_p metrics for convex bodies | math.MG | Intrinsic $L_p$ metrics are defined and shown to satisfy a dimension--free
bound with respect to the Hausdorff metric. | math |
1,977 | Hyperbolic Coxeter n-polytopes with n+2 facets | math.MG | In this paper, we classify all the hyperbolic non-compact Coxeter polytopes
of finite volume combinatorial type of which is either a pyramid over a product
of two simplices or a product of two simplices of dimension greater than one.
Combined with results of Kaplinskaja (1974) and Esselmann (1996) this completes
the cl... | math |
1,978 | The multiplicative structure on polynomial continuous valuations | math.MG | We introduce a canonical structure of a commutative associative filtered
algebra with the unit on polynomial smooth valuations, and study its
properties. The induced structure on the subalgebra of translation invariant
smooth valuations has especially nice properties (it is the structure of the
Frobenius algebra). We a... | math |
1,979 | Beckman-Quarles type theorems for mappings from R^n to C^n | math.MG | Let G: C^n \times C^n -> C, G((x_1,...,x_n),(y_1,...,y_n))=(x_1-y_1)^2+...+
(x_n-y_n)^2. We say that f: R^n -> C^n preserves distance d>0 if for each x,y
\in R^n G(x,y)=d^2 implies G(f(x),f(y))=d^2. Let A(n) denote the set of all
positive numbers d such that any map f: R^n -> C^n that preserves unit distance
preserves ... | math |
1,980 | Total curvature and spiralling shortest paths | math.MG | This paper gives a partial confirmation of a conjecture of P. Agarwal, S.
Har-Peled, M. Sharir, and K. Varadarajan that the total curvature of a shortest
path on the boundary of a convex polyhedron in the 3-dimensional Euclidean
space cannot be arbitrarily large. It is shown here that the conjecture holds
for a class o... | math |
1,981 | Singularities of convex hulls of smooth hypersurfaces | math.MG | We describe singularities of the convex hull of a generic compact smooth
hypersurface in four-dimensional affine space up to diffeomorphisms. It turns
out there are only two new singularities (in comparison with the previous
dimension case) which appear at separate points of the boundary of the convex
hull and are not ... | math |
1,982 | A discrete form of the Beckman-Quarles theorem for mappings from R^2 (C^2) to F^2, where F is a subfield of a commutative field extending R (C) | math.MG | Let F be a subfield of a commutative field extending R. Let phi_n:F^n \times
F^n ->F, phi_n((x_1,...,x_n),(y_1,...,y_n))=(x_1-y_1)^2+...+(x_n-y_n)^2. We say
that f:R^n->F^n preserves distance d>=0 if for each x,y \in R^n |x-y|=d implies
phi_n(f(x),f(y))=d^2. Let A_n(F) denote the set of all positive numbers d such
that... | math |
1,983 | Pinched exponential volume growth implies an infinite dimensional isoperimetric inequality | math.MG | Let $G$ be a graph which satisfies $c^{-1} a^r \le |B(v,r)| \le c a^r$, for
some constants $c,a>1$, every vertex $v$ and every radius $r$. We prove that
this implies the isoperimetric inequality $|\partial A| \ge C |A| / \log(2+
|A|)$ for some constant $C=C(a,c)$ and every finite set of vertices $A$. | math |
1,984 | Lattice packings with gap defects are not completely saturated | math.MG | We show that a honeycomb circle packing in $\R^2$ with a linear gap defect
cannot be completely saturated, no matter how narrow the gap is. The result is
motivated by an open problem of G. Fejes T\'oth, G. Kuperberg, and W.
Kuperberg, which asks whether of a honeycomb circle packing with a linear shift
defect is comple... | math |
1,985 | Algebraic structures on valuations, their properties and applications | math.MG | We describe various structures of algebraic nature on the space of continuous
valuations on convex sets, their properties (like versions of Poincar\'e
duality and hard Lefschetz theorem), and their relations and applications to
integral geometry. | math |
1,986 | A computer verification of the Kepler conjecture | math.MG | The Kepler conjecture asserts that the density of a packing of congruent
balls in three dimensions is never greater than $\pi/\sqrt{18}$. A computer
assisted verification confirmed this conjecture in 1998. This article gives a
historical introduction to the problem. It describes the procedure that
converts this problem... | math |
1,987 | Mappings from R^n to F^n which preserve unit Euclidean distance, where F is a field of characteristic 0 | math.MG | Let F be a commutative field of characteristic 0, G_n: F^n \times F^n -> F,
G_n((x_1,...,x_n),(y_1,...,y_n))=(x_1-y_1)^2+...+(x_n-y_n)^2. We say that
g:R^n->F^n preserves distance d>=0 if for each x,y \in R^n |x-y|=d implies
G_n(g(x),g(y))=d^2. Let f:R^n->F^n preserve unit distance. We prove: (1) if
n>=2, x,y \in R^n a... | math |
1,988 | The Beckman-Quarles theorem for mappings from R^2 to F^2, where F is a subfield of a commutative field extending R | math.MG | Let F be a subfield of a commutative field extending R. Let \phi_2: F^2
\times F^2 \to F, \phi_2((x_1,x_2),(y_1,y_2))=(x_1-y_1)^2+(x_2-y_2)^2. We say
that f:R^2 \to F^2 preserves distance d \geq 0 if for each x,y \in R^2 |x-y|=d
implies \phi_2(f(x),f(y))=d^2. We prove that each unit-distance preserving
mapping f:R^2 \t... | math |
1,989 | omega-Periodic graphs | math.MG | $\omega$-periodic graphs are introduced and studied. These are graphs which
arise as the limits of periodic extensions of the nearest neighbor graph on the
integers. We observe that all bounded degree $\omega$-periodic graphs are
ameanable. We also provide examples of $\omega$-periodic graphs which have
exponential vol... | math |
1,990 | Free planes in lattice sphere packings | math.MG | We show that for every lattice packing of $n$-dimensional spheres there
exists an $(n/\log_2(n))$-dimensional affine plane which does not meet any of
the spheres in their interior, provided $n$ is large enough. Such an affine
plane is called a free plane and our result improves on former bounds. | math |
1,991 | From spaces of polygons to spaces of polyhedra following Bavard, Ghys and Thurston | math.MG | After work of W. P. Thurston, C. Bavard and \'E. Ghys constructed particular
hyperbolic polyhedra from spaces of deformations of Euclidean polygons. We
present this construction as a straightforward consequence of the theory of
mixed-volumes. The gluing of these polyhedra can be isometrically embedded into
complex hype... | math |
1,992 | Some observations on the simplex | math.MG | We investigate the space of simplices in Euclidean Space | math |
1,993 | The Double Bubble Problem on the Flat Two-Torus | math.MG | We characterize the perimeter-minimizing double bubbles on all flat two-tori
and, as corollaries, on the flat infinite cylinder and the flat infinite strip
with free boundary. Specifically, we show that there are five distinct types of
minimizers on flat two-tori, depending on the areas to be enclosed. | math |
1,994 | On locally convex PL-manifolds and fast verification of convexity | math.MG | We show that a realization of a closed connected PL-manifold of dimension n-1
in Euclidean n-space (n>2) is the boundary of a convex polyhedron if and only
if the interior of each (n-3)-face has a point, which has a neighborhood lying
on the boundary of a convex n-dimensional body. This result is derived from a
general... | math |
1,995 | The kissing number in four dimensions | math.MG | The kissing number problem asks for the maximal number k(n) of equal size
nonoverlapping spheres in n-dimensional space that can touch another sphere of
the same size. This problem in dimension three was the subject of a famous
discussion between Isaac Newton and David Gregory in 1694. In three dimensions
the problem w... | math |
1,996 | Some Properties of Lattice Substitution Systems | math.MG | If a partition of a lattice in R^d is selfsimilar, it is called lattice
substitution system (LSS). Such sets represent nonperiodic, but highly ordered
structures. An important property of such structures is, whether they are model
sets or not (equivalently, whether they are pure point diffractive or not). In
this paper... | math |
1,997 | Hyperbolic Coxeter n-polytopes with n+3 facets | math.MG | A polytope is called a Coxeter polytope if its dihedral angles are integer
parts of $\pi$. In this paper we prove that if a non-compact Coxeter polytope
of finite volume in $H^n$ has exactly $n+3$ facets then $n\le 16$. We also find
an example in $H^{16}$ and show that it is unique. | math |
1,998 | Sphere Packings in Hyperbolic Space: Periodicity and Continuity | math.MG | This paper is being withdrawn because an error was discovered in lemma 4.3.
Although the rest of the paper appears to be correct, this error invalidates
the proof of theorem 3.1 and theorem 3.3. | math |
1,999 | Curvature of sub-Riemannian spaces | math.MG | To any metric spaces there is an associated metric profile. The
rectifiability of the metric profile gives a good notion of curvature of a
sub-Riemannian space. We shall say that a curvature class is the rectifiability
class of the metric profile. We classify then the curvatures by looking to
homogeneous metric spaces.... | math |
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