Helly theorem application

Question

Let $X$ be a normed space, $dim(X)=d$, let $r>0, A$ and $\subset X$ . Show that if every $d+1$ points of $A$ are contained in a closed ball of radius $r$, then $A$ is contained in a closed ball of radius $r$. After immerging $X$ in $R^d$,my idea is using Helly theorem, I have to show that the family of balls is a $d+1$ centered family, it seems simple but i can't. Then the sets are convex, their intersection is non empty, so I'm done i think! Any suggestion? Thanks.

Answer 1

For each $x\in X$ let $C_x$ be the closed ball of radius $r$ centred at $x$; note that $x\in C_y$ if and only if $\|x-y\|\le r$ if and only if $y\in C_x$.

If $x_1,\ldots,x_{d+1}\in A$, there is an $x\in X$ such that $\{x_1,\ldots,x_{d+1}\}\subseteq C_x$, and it follows from the observation in the first paragraph that $x\in\bigcap_{k=1}^{d+1}C_{x_k}\ne\varnothing$.

(You were already using the observation in the first paragraph to go from $\bigcap_{x\in A}C_x\ne\varnothing$ to the existence of an $x\in X$ such that $A\subseteq C_x$!)