The extreme points of $\overline{\mathrm{conv}(A)}$ are in $\overline{A}$

Question

Let $A$ be a subset of $\mathbb{R}^n$ and let $e$ be an extreme point of $\overline{\mathrm{conv}(A)}$. Prove that $e \in \overline{A}$.

The problem is intuitive, but I cannot prove it. Thanks in advance for any help!

Answer 1

Suppose that $p\not\in\overline{A}$ and take some $r>0$ such that the ball $B=B(p,r)$ misses $\overline{A}$. Assume also that $p\in\overline{\mathrm{conv}(A)}$. Let $S$ be the boundary of $B$, that is $S=\{x:||x-p||=r\}$. Using compactness of $S$ we are going to show that $p$ is not an extreme point of $\overline{\mathrm{conv}(A)}$. (This shows that if $e$ is an extreme point of $\overline{\mathrm{conv}(A)}$ then $e\in\overline{A}$.)

Take a sequence of points $p_k$ in $B$ converging to $p$ as $k\to\infty$, and such that $p_k\in\mathrm{conv}(A)$ for each $k$. For each $k$ pick a finite set $F_k\subseteq A$ such that $p_k\in\mathrm{conv}(F_k)$. Since we work in $\Bbb R^n$ we may assume that each $F_k$ has at most $n+1$ many points. Then there is some $m\le n+1$ such that there are infinitely many $k$ such that $F_k$ has exactly $m$ many points. By passing to a subsequence (throwing out the remaining $k$'s and respective $p_k$'s) we may assume that $F_k$ has exactly $m$ many points for every $k$, say $F_k=\{x_{k,j}:1\le j\le m\}$. Write $p_k= \lambda_{k,1}x_{k,1}+...+\lambda_{k,m}x_{k,m}$, a convex combination with suitable $\lambda_{k,1},...,\lambda_{k,m}$.

For each $k$ and each $j$ with $1\le j\le m$ let $s_{k,j}$ be the intersection with $S$ of the line segment joining $p_k$ and $x_{k,j}$. If $S_k=\{s_{k,j}:1\le j\le m\}$ then $S_k\subseteq\mathrm{conv}(A)$ (since each $s_{k,j}$ is a convex combination of $p_k\in\mathrm{conv}(A)$ and $x_{k,j}\in A$).

Lemma. $p_k=\mu_{k,1}s_{k,1}+...+\mu_{k,m}s_{k,m}$ a convex combination with suitable $\mu_{k,1},...,\mu_{k,m}$.
Proof. There are some $0<t_{k,j}<1$ for $1\le j\le m$ such that $s_{k,j}=(1-t_{k,j})p_k+t_{k,j}x_{k,j}$. Then $x_{k,j}=\frac{s_{k,j}-(1-t_{k,j})p_k}{t_{k,j}}$, hence $p_k= \lambda_{k,1}x_{k,1}+...+\lambda_{k,m}x_{k,m}=$ $\lambda_{k,1}\frac{s_{k,1}-(1-t_{k,1})p_k}{t_{k,1}}+... +\lambda_{k,m}\frac{s_{k,m}-(1-t_{k,m})p_k}{t_{k,m}}$. Solving for $p_k$ we obtain
$p_k[1+\lambda_{k,1}\frac{(1-t_{k,1})}{t_{k,1}}+... +\lambda_{k,m}\frac{(1-t_{k,m})}{t_{k,m}}]= \frac{\lambda_{k,1}}{t_{k,1}}s_{k,1}+... +\frac{\lambda_{k,m}}{t_{k,m}}s_{k,m}$, and
$p_k[1+\frac{\lambda_{k,1}}{t_{k,1}}-\lambda_{k,1}+... +\frac{\lambda_{k,m}}{t_{k,m}}-\lambda_{k,m}]= p_k[\frac{\lambda_{k,1}}{t_{k,1}}+... +\frac{\lambda_{k,m}}{t_{k,m}}] = \frac{\lambda_{k,1}}{t_{k,1}}s_{k,1}+... +\frac{\lambda_{k,m}}{t_{k,m}}s_{k,m}$.
Thus $p_k =[\frac{\lambda_{k,1}}{t_{k,1}}s_{k,1}+... +\frac{\lambda_{k,m}}{t_{k,m}}s_{k,m}]/[\frac{\lambda_{k,1}}{t_{k,1}}+... +\frac{\lambda_{k,m}}{t_{k,m}}]$. That is, coefficients $\mu_{k,j}=[\frac{\lambda_{k,j}}{t_{k,j}}]/[\frac{\lambda_{k,1}}{t_{k,1}}+... +\frac{\lambda_{k,m}}{t_{k,m}}]$ for $1\le j\le m$ work.

We may pick convergent subsequences several times (throwing out some $k$'s each time), first so that we may assume that the sequence $\{s_{k,1}:k\ge1\}$ is convergent to some $s_1\in S$, then we may assume that the sequence $\{\mu_{k,1}:k\ge1\}$ is convergent to some $\mu_1\in[0,1]$, then we may assume that the sequence $\{s_{k,2}:k\ge1\}$ is convergent to some $s_2\in S$, etc. In the end we may assume that for each fixed $j$ with $1\le j\le m$ the sequence $\{s_{k,j}:k\ge1\}$ is convergent to some $s_j\in S$, and the sequence $\{\mu_{k,j}:k\ge1\}$ is convergent to some $\mu_j\in[0,1]$. Since $p_k\to p$ we also have that $p=\mu_1s_1+...+\mu_m s_m\ ,$ a convex combination. If $T=\{s_1,...,s_m\}$ then $T\subset S\cap\overline{\mathrm{conv}(A)}$, and in particular $p\in\mathrm{conv}(T)\subset\overline{\mathrm{conv}(A)}$ (using that $\overline{\mathrm{conv}(A)}$ is convex). But $p$ is not an extreme point of $\mathrm{conv}(T)$ (and therefore $p$ cannot be an extreme point of $\overline{\mathrm{conv}(A)}$ either), which could be shown by induction on $m$. If $m=2$ then $p$ is an interior point of the line segment joining $s_1$ and $s_2$. If $m=3$ and $p$ does not lie on the line segment joining $s_2$ and $s_3$ then there is a point $q$ on this line segment so that $p$ is an interior point of the line segment joining $s_1$ and $q$. Etc, for $m\ge4$. (Or, we may just use that since $T$ is finite, all extreme points of $\mathrm{conv}(T)$ must be contained in $T$, and therefore in $S$, but $p\not\in S$. It could also be seen, though it is not needed for the proof, that $T$ is exactly the set of extreme points of $\mathrm{conv}(T)$, using that each point of $S$ is an extreme point for $\mathrm{conv}(S)$.)