Convex sets

Prove that every flat of an affine space is the intersection of a finite number of half-spaces.

Prove that a subset of a flat $P$ of an affine space $A$ is a convex polytope in $P$ if and only if it is a convex polytope in $A$.

Let a convex polytope $M$ in an affine space be given by a system of linear inequalities

Prove that for every nonempty subset $J \subseteq \left\{ 1, \ldots , k\right\}$ the set $M^{J}$, defined by the conditions $f_{i}(x)=0$ for $i \in J, f_{i}(x) \geqslant 0$ for $i \notin J$, if nonempty, is a face of the polytope $M$, and, conversely, every face of the polytope $M$ has the form $M^{J}$ for some set $J \subseteq \left\{ 1, \ldots , k\right\}$.

Let $a_{0}, a_{1}, \ldots , a_{n}$ be points of an $n$-dimensional affine space in general position, $H_{i}(i=0,1, \ldots , n)$ the hyperplane passing through all these points except $a_{i}$, and $H_{i}^{+}$ the half-space bounded by it that contains the point $a_{i}$. Prove that

Prove that the faces of an $n$-dimensional simplex

are the convex hulls of all possible proper subsets of the set $\left\{ a_{0}, a_{1}, \ldots , a_{n}\right\}$.

Find the faces of an $n$-dimensional parallelepiped given, in some system of affine coordinates, by the inequalities $0 \leqslant x_{i} \leqslant 1(i=1$, $2, \ldots , n)$.

Find the vertices and describe the shape of the convex polytope in a three-dimensional affine space given by the inequalities

Find the vertices and describe the shape of the sections of the four-dimensional parallelepiped given by the inequalities $0 \leqslant x_{i} \leqslant 1(i=1,2,3,4)$, by the hyperplanes:

Prove that the closure of a convex set is convex.

Prove that the open kernel $M^{0}$ of a solid convex set $M$ is convex and its closure contains $M$.

Prove that the image and the full preimage of a convex set under an affine map are convex sets.

Prove that:

Prove that the convex hull of a set $S$ consists of all possible combinations of the form $\sum_{i=1}^{k} \lambda_{i} a_{i}$, where

Let $M$ be a convex set and $a \notin M$. Prove that

Let $S$ be a subset of an $n$-dimensional affine space $A$. Prove that if $\langle S\rangle =A$, then conv $S$ is a union of $n$-dimensional simplices with vertices at points of the set $S$.

Prove that the convex hull of a compact set is compact.

Let $M$ be a convex subset of a two-dimensional affine space and $a \notin M^{0}$ (see 50.10). Prove that a line can be drawn through the point $a$ such that the set $M$ lies on one side of it.

Let $M$ be a convex subset of an $n$-dimensional affine space $A$ and $a \notin M^{0}$ (see 50.10). Prove that a hyperplane can be drawn through the point $a$ such that the set $M$ lies on one side of it.

Prove that through any point of a closed convex set not belonging to its open kernel, a supporting hyperplane can be drawn.

Prove that every closed convex set $M$ is the intersection of (in general, infinitely many) half-spaces.

Prove that every closed convex cone in a vector space is the intersection of (in general, infinitely many) half-spaces whose boundaries pass through zero.

Let $f_{i}(i=1, \ldots , k)$ be affine linear functions on an affine space $A$. Prove that the system of inequalities $f_{i}(x) \leqslant 0(i=1, \ldots , k)$ is inconsistent if and only if there exist numbers $\lambda_{i} \geqslant 0$ such that $\sum_{i=1}^{k} \lambda_{i} f_{i}$ is a positive constant.

Let $M$ be a compact convex set containing a neighborhood of zero, in a vector space $V$ regarded as an affine space, and let

Prove that:

Prove that every compact convex set coincides with the convex hull of the set of its extreme points.

Prove that the maximum of an affine linear function on a compact convex set is attained at some extreme point (but may also be attained at other points).

Prove that the extreme points of a convex polytope are its vertices.

Prove that every bounded convex polytope coincides with the convex hull of the set of its vertices.

Prove that the convex hull of a finite number of points is a convex polytope.

Give, by a system of linear inequalities, the convex hull of the indicated points of a four-dimensional affine space, and find the three-dimensional faces of this convex polytope:

Let $M$ and $N$ be convex sets in an affine space $(A, V)$. Prove that:

Let $M$ and $N$ be disjoint closed convex sets in an affine space $A$, one of which is bounded. Prove that there exists an affine linear function $f$ on $A$ such that $f(x)<0$ for all $x \in M$ and $f(y)>0$ for all $y \in N$.

Let $M$ be a compact convex set in an affine space $A$, $N$ a compact convex set in the vector space $L$ of all affine linear functions on $A$, and suppose that for every point $a \in M$ there is a function $f \in N$ such that $f(a) \geqslant 0$. Prove that there exists a function $f_{0} \in N$ such that $f_{0}(x) \geqslant 0$ for all $x \in M$.

Let $F$ be an affine bilinear function on the direct product of affine spaces $A$ and $B$, and let $M$ and $N$ be compact convex subsets of the spaces $A$ and $B$, respectively.

Prove that:

Prove that:

Determine whether the polytopes given by the following inequalities are bounded:

Find the vertex points of the polytopes:

Find the maximum and minimum values of the linear function $z$ on a bounded polytope: