So I'm struggling with this exercise from Lay's Convex Sets:
Let $\mathcal{F}$ be a family of compact convex sets in $\mathbb R^n$ containing at least $n+1$ members. Suppose that for each subfamily of $n+1$ members there exists a point $x$ such that all sets in the subfamily are at a distance less or equal than $d \ge 0$ from $x$. Then there exists $y$ such that all sets in $\mathcal F$ are at a distance $ \le d$ from $y$.
I feel like I should use Helly's theorem, but it is not clear to me how to.
Thanks for any help.
Indeed, this is typical question amendable by Helly's theorem.
Say $\mathcal F = (F_j)_{j \in J}$, and let $d \ge 0$ be as in the announcement. It's clear that for each $j \in J$, the set $X_j := \{x \in \mathbb R^n| \mathrm{dist}(F_j, x) \le d\}$ is convex and compact. Now for any $n + 1$ indices $j_1, \ldots, j_{n+1} \in J$, the hypothesis ensures that there is a point $x$ which is at distance $\le d$ from any $F_{j_k}$, i.e $x \in \cap_{1 \le k \le n + 1}X_{j_k}$. But then Helly's theorem implies $\cap_{j \in J}X_j \ne \emptyset$, i.e there exists a point which is at a distance $\le d$ from any $F \in \mathcal F$.