# Exercises 4

\centerline{\bf CONVEX SET}
\bigskip
\subsection{Problem 1}
Prove that section A$[a,b]\subset \Bbb R^n$ is a convex set.
\subsection{Problem 2}
Specify all the nonzero convex subsets of $\Bbb R$.
\subsection{Problem 3}
A convex polygon lies on a plane. It does not lie on one line
and lines that join to point of the polygon is a side of the
polygon. Prove that the polygon is a triangle.
\subsection{Problem 4}
Point a lies on the face $F$ of a convex set $X$ and is
a convex combination of points $a_1,\dots, a_k\in X$ with nonzero
coefficients. Prove that $a_1, \dots,a_k\in F$.
\subsection{Problem 5}
Prove that a convex polygon in $\Bbb R^3$ any two vertexes
of which are joined by an edge is a tetrahedron.
\subsection{Problem 6 {\rm (Radon's theorem on plane)}}
On a plain there are given $m\geq 4$ points $a_1,\dots, a_m$.
Prove that they can be divided into two groups, convex hulls of
which intersect.
\subsection{Problem 7}
Suppose that $a_1, \dots, a_m\in \Bbb R^n$, $m\geq n+2$. Prove
that there exist $\lambda_1\dots, \lambda_m$ not all of them zero
that $\lambda_1a_1+ \dots +\lambda_m a_m=0$, $\lambda_1+\dots +\lambda_m=0$.
\subsection{Problem 8 {\rm (Radon's theorem)}}
Suppose that $a_1, \dots, a_m\in \Bbb R^n$, $m>\geq n+2$. Prove
that they can be divided into two groups, convex hulls of which
intersect.
\subsection{Problem 9 {\rm (Helly's theorem)}}
Suppose $M_1, \dots, M_m\subset \Bbb R^n$ - convex sets, $m \geq n+1$. It is given that the intersection of any $n+1$ set $M_i$ is
nonzero. Prove that the intersection of all of the $M_i$ is
nonzero.
\subsection{Problem 10}
Suppose that $T_1,\dots, T_m$ -- parallel sections on a plane, $m\geq 3$.
It is given that for any three sections there exits a line that
intersects all of them. Prove that there exits a line that
intersects all of the sections.