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If $f$ is convex on the interval $[x,y]$, then for any $c \in [x,y]$, we have $f(c) \leq \frac{f(x) - f(y)}{y-x}(y-c) + f(y)$. |
A translation of a convex set is convex. |
The set $s$ is convex if and only if the set $b - a$ is convex. |
If $f$ is a linear injection, then $f(S)$ is convex if and only if $S$ is convex. |
The function $(x,y) \mapsto x + y$ is linear. |
For any nonnegative real number $c$, there exists a vector $x$ of norm $c$. |
For every $c \geq 0$, there exists a vector $y$ such that $||x - y|| = c$. |
If $s$ is a finite set, then $\sum_{x \in s} (\text{if } y = x \text{ then } f(x) \text{ else } 0) \cdot x = (\text{if } y \in s \text{ then } f(y) \cdot y \text{ else } 0)$. |
For any three points $x, y, z$ in a real inner product space, the following are equivalent: $\|x - z\| = \|x - y\| + \|y - z\|$ $\langle x - y, y - z \rangle = \|y - z\|^2$ |
The cone of the empty set is the empty set. |
The cone generated by the empty set is the entire space. |
If $s$ is a cone for every $s \in f$, then $\bigcap f$ is a cone. |
If $S$ is a subspace, then it is a cone. |
The cone hull of a set is a cone. |
The cone hull of a set $S$ is equal to $S$ if and only if $S$ is a cone. |
If $S$ is a cone and $x \in S$, then $cx \in S$ for all $c \geq 0$. |
A cone is nonempty if and only if it contains the origin. |
The set $\{0\}$ is a cone. |
If every element of a set $f$ is a cone, then the union of $f$ is a cone. |
A set $S$ is a cone if and only if $0 \in S$ and for all $c > 0$, $cS = S$. |
The cone hull of the empty set is the empty set. |
The cone hull of an empty set is empty. |
The cone hull of a set $S$ is nonempty if and only if $0 \<in> S$. |
If $x$ is in a set $S$ and $c$ is a nonnegative real number, then $cx$ is in the cone hull of $S$. |
The cone hull of a set $S$ is the set of all nonnegative linear combinations of elements of $S$. |
A set $S$ is a convex cone if and only if for all $x, y \in S$, $x + y \in S$ and for all $x \in S$ and $c \geq 0$, $cx \in S$. |
If $S$ is a convex set, then $S$ is connected. |
The real vector space $\mathbb{R}^n$ is connected. |
If $P_i$ is a convex set for each $i \in \{1, \ldots, n\}$, then the set $\{(x_1, \ldots, x_n) \in \mathbb{R}^n \mid x_i \in P_i \text{ for all } i\}$ is convex. |
The set of points in Euclidean space whose coordinates are all nonnegative is convex. |
The convex hull of a set is convex. |
If $s$ is a subset of the convex hull of $t$, then the convex hull of $s$ is a subset of the convex hull of $t$. |
The convex hull of a set $s$ is equal to $s$ if and only if $s$ is convex. |
If $f$ is a linear map, then $f$ maps the convex hull of a set $S$ to the convex hull of $f(S)$. |
If $f$ is a linear map and $x$ is in the convex hull of $s$, then $f(x)$ is in the convex hull of $f(s)$. |
The convex hull of the Cartesian product of two sets is the Cartesian product of the convex hulls of the two sets. |
The convex hull of the empty set is the empty set. |
The convex hull of a single point is the point itself. |
The convex hull of a set $S$ and a point $a$ is the set of all points that can be written as a convex combination of $a$ and a point in $S$. |
The convex hull of a set $S$ and a point $a$ is the set of all points that can be written as a convex combination of $a$ and a point in $S$. |
The convex hull of a set $S$ is the set of all points that can be written as a convex combination of points in $S$. |
If $S$ is a finite set, then the convex hull of $S$ is the set of all convex combinations of elements of $S$. |
The convex hull of a set $p$ is the set of all points that can be written as a convex combination of points in $p$. |
If $S$ is a finite set, then $y \in \text{conv}(S)$ if and only if there exists $v \geq 0$ and $u$ such that $\sum_{x \in S} u(x) = w - v$ and $\sum_{x \in S} u(x) x = y - v a$. |
The convex hull of two points $a$ and $b$ is the set of all points of the form $u a + v b$ where $u$ and $v$ are non-negative real numbers that sum to $1$. |
The convex hull of two points is the line segment between them. |
The convex hull of three points $a$, $b$, and $c$ is the set of all points of the form $u a + v b + w c$, where $u$, $v$, and $w$ are nonnegative real numbers that sum to $1$. |
The convex hull of three points $a$, $b$, and $c$ is the set of all points of the form $a + u(b - a) + v(c - a)$ where $u$ and $v$ are nonnegative real numbers that sum to at most $1$. |
Any affine set is convex. |
The affine hull of a set is convex. |
Any subspace of a vector space is convex. |
The convex hull of a set $s$ is a subset of the span of $s$. |
The convex hull of a set is contained in its affine hull. |
The affine dimension of the convex hull of a set is equal to the affine dimension of the set. |
The convex hull of a set $p$ is the set of all points that can be written as a convex combination of at most $d+1$ points of $p$, where $d$ is the affine dimension of $p$. |
The convex hull of a set $p$ is the set of all points that can be written as a convex combination of at most $n+1$ points in $p$, where $n$ is the affine dimension of $p$. |
The convex hull of a set $p$ is the set of all points that can be written as a convex combination of at most $n+1$ points of $p$, where $n$ is the dimension of the space. |
The convex hull of a set $p$ is the set of all points $x$ that can be written as a convex combination of at most $n + 1$ points in $p$, where $n$ is the dimension of the space. |
If $d$ is a subset of the basis of $\mathbb{R}^n$, then the affine hull of $d$ is the set of all vectors in $\mathbb{R}^n$ whose coordinates are zero outside of $d$. |
The affine hull of the convex hull of a set $S$ is the affine hull of $S$. |
The convex hull of the sum of two sets is the sum of the convex hulls of the two sets. |
The image of a set $T$ under the translation map $x \mapsto a + x$ is equal to the set $a + T$. |
The convex hull of a translation of a set is the translation of the convex hull of the set. |
The convex hull of a set $S$ is equal to the set of all convex combinations of points in $S$. |
The convex hull of the image of a set $S$ under an affine transformation is the image of the convex hull of $S$ under the same affine transformation. |
If $S$ is a convex set, then the convex cone generated by $S$ is convex. |
If $S$ is a cone, then the convex hull of $S$ is also a cone. |
If $c$ is a finite affinely dependent set, then there exists a function $u$ from $c$ to $\mathbb{R}$ such that $\sum_{v \in c} u(v) = 0$, $u(v) \neq 0$ for some $v \in c$, and $\sum_{v \in c} u(v) v = 0$. |
If $S$ is a finite set and $\sum_{x \in S} f(x) = 0$, then $\sum_{x \in S, f(x) > 0} f(x) = - \sum_{x \in S, f(x) < 0} f(x)$. |
If $S$ is a finite set, $f$ is a function from $S$ to $\mathbb{R}^n$, and $g$ is a function from $S$ to $\mathbb{R}$, then $\sum_{x \in S} f(x) = 0$ if and only if $\sum_{x \in S, g(x) > 0} f(x) = - \sum_{x \in S, g(x) < 0} f(x)$. |
If $C$ is a finite affinely dependent set, then there exist two disjoint subsets $m$ and $p$ of $C$ such that $m \cup p = C$ and the convex hulls of $m$ and $p$ intersect. |
If a set of points in the plane is affinely dependent, then there are two disjoint subsets of the set whose convex hulls intersect. |
If $f$ is a finite collection of convex sets in $\mathbb{R}^n$ such that every $n+1$ of them have a point in common, then all of them have a point in common. |
Helly's theorem: If $f$ is a finite family of convex sets in $\mathbb{R}^n$ such that every $n+1$ of them have a nonempty intersection, then $\bigcap f$ is nonempty. |
$(x, y) \in \text{epigraph}(S, f)$ if and only if $x \in S$ and $f(x) \leq y$. |
The epigraph of a convex function is convex if and only if the function is convex and the domain is convex. |
If $f$ is convex on $S$ and $S$ is convex, then the epigraph of $f$ is convex. |
If $S$ is convex and $f$ is convex on $S$, then the epigraph of $f$ is convex. |
A function $f$ is convex on a convex set $S$ if and only if for all $k \in \mathbb{N}$, $u \in \mathbb{R}^k$, and $x \in S^k$, if $\sum_{i=1}^k u_i = 1$ and $x_i \in S$ for all $i$, then $f(\sum_{i=1}^k u_i x_i) \leq \sum_{i=1}^k u_i f(x_i)$. |
If $f$ is convex on the convex hull of $S$ and $f(x) \leq b$ for all $x \in S$, then $f(x) \leq b$ for all $x$ in the convex hull of $S$. |
If $i$ is a basis vector, then $\sum_{j \in \text{Basis}} j \cdot i = 1$. |
If $S$ and $T$ are convex sets, then $S + T$ is convex. |
If each $B_i$ is convex, then $\sum_{i \in A} B_i$ is convex. |
If $A$ is a finite set and for each $i \in A$, $B_i$ is a finite set, then $\bigcup_{i \in A} B_i$ is a finite set. |
The set of points in $\mathbb{R}^n$ whose $i$th coordinate is in $B_i$ is equal to the set of points in $\mathbb{R}^n$ whose $i$th coordinate is in $B_i$ for all $i$. |
The convex hull of the sum of a family of sets is the sum of the convex hulls of the sets. |
$y \in B(x, e)$ if and only if $d(x, y) < e$. |
A point $y$ is in the closed ball of radius $e$ centered at $x$ if and only if the distance between $x$ and $y$ is less than or equal to $e$. |
A point $y$ is on the sphere of radius $e$ centered at $x$ if and only if the distance between $x$ and $y$ is $e$. |
The ball of radius $0$ around any point is empty. |
The closed ball of radius $0$ around $x$ is the singleton set $\{x\}$. |
The sphere of radius $0$ around $x$ is just the point $x$. |
If the distance between two points $x$ and $y$ is greater than the sum of the radii of two balls centered at $x$ and $y$, then the two balls are disjoint. |
If the distance between $x$ and $y$ is greater than $r + s$, then the closed balls of radius $r$ and $s$ centered at $x$ and $y$, respectively, are disjoint. |
The sphere of radius $r$ around a point $a$ is empty if $r < 0$. |
The centre of a ball is in the ball if and only if the radius is positive. |
The centre of a ball is in the ball if and only if the radius is non-negative. |
The open ball of radius $e$ centered at $x$ is contained in the closed ball of radius $e$ centered at $x$. |
If $x$ is in the open ball of radius $e$ centered at $y$, then $x$ is in the closed ball of radius $e$ centered at $y$. |
The sphere of radius $r$ centered at $z$ is contained in the closed ball of radius $r$ centered at $z$. |
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