I was reading a book that gathered several interesting math results and I came accross a theorem from Helly
Theorem (Helly) : Let $m, n$ be two integers such that $m \geq n+1 \geq 2$. Let be $K_1, \ldots, K_m$ be $m$ convex compacts subsets of $\mathbb R^n$ and assume that every intersection of $n+1$ of those sets is non-empty. Then $\bigcap_{1 \leq i \leq m} K_i$ is non-empty.
This is a well-known result in convex sets. However, it's written in the book that an application of this theorem is the following result, attributed to Krasnoel'skii
Proposition (Krasnoel'skii) : Let $K$ be a compact subset of $\mathbb R^n$ such that for every $x_1, \ldots, x_{n+1} \in K$, there exists $x \in K$ such that $[x, x_i] \subset K$ for all $1 \leq i \leq n+1$. Then $K$ is a star domain (there exists $x_0 \in K$ such that for every $x \in K$, $[x_0, x] \subset K$).
(it reminds a bit of the Art gallery problem)
My idea in order to prove Krasnoel'skii's result was to use the compactness of $K$ to get a finite open-cover $\displaystyle K \subset \bigcup_{i=1}^m B(x_i, r_i)$ for some $x_i \in K$ and $r_i > 0$. If $m < n+1$ we can always add some balls to this cover, so we may assume $m \geq n+1$. Thanks to the hypothesis on $K$, for every subset $J \subset \lbrace 1, \ldots, m \rbrace$ of size $n+1$, there exists $x_J \in K$ such that $[x_J, x_i] \subset K$ for all $i \in J$. Therefore, the compact sets $K_i := \bigcup_{J \subset \lbrace 1, \ldots, m \rbrace, |J| = n+1} [x_J, x_i]$ (for every $i = 1, 2, \ldots, m$) satisfy the condition of Helly's theorem and have a non-empty intersection, say $y$. I guess $y$ is a good candidate for $K$ to be stared at $y$, but I cannot figure it out. The segments $[y, x]$ for $x \in K \backslash \bigcup_i K_i$ have no reason to lie in $K$.
In fact, I feel that we cannot show more than that, but I might be wrong. And the book doesn't give any reference for this statement... There might be some subtle thinking to do whith the radii $r_i$ of the balls, maybe taking $y_{\epsilon}$ (instead of $y$) that depends on the maximum of the radii $r_i$ and take $\epsilon \to 0$, but I am unsure about how it would work out. Do you have any ideas ?