Let's $K\subset\mathbb{R}^n$ compact. If $K$ is convex then who to prove that any point of $K$ is convex combination of one or two extremal points of $K$?
Intuitively, for any closed ball that is true.
Convex Analysis – Convex Combination in Compact Convex Sets
convex-analysis
Best Answer
The fact that in ${\mathbb R}^n$ each point of a compact convex set is a convex combination of at most $n+1$ extreme points is a theorem of Carathéodory. You can prove this by induction on $n$.
The case $n=0$ is easy. For the induction step, if $K$ is a compact convex set in ${\mathbb R}^{n+1}$ and $x \in K$, choosing some extreme point $y$ of $K$ we have $x = t y + (1-t) z$ where $0 \le t \le 1$ and $z$ is a boundary point of $K$. $K$ has a supporting hyperplane $H$ at $z$, and $H \cap K$ is a compact convex set in the $n$-dimensional space $H$ whose extreme points are extreme points of $K$. So represent $z$ as a convex combination $z = \sum_{i=1}^{n+1} c_i z_i$ of at most $n+1$ of these extreme points, and $x = t y + \sum_{i=1}^{n+1} (1-t) c_i z_i$ is a convex combination of at most $n+2$ extreme points of $K$.