TITLE: covering convex sets by round balls
QUESTION [9 upvotes]: Let $X=\{x_1,...,x_k\}\subset E^n$ be a finite subset in the Euclidean $n$-space, $r>0$ and $B(x_i,r)$ are open balls of radius $r$ centered at the points $x_i\in X, i=1,...,k$. Suppose that
$$
\bigcap_{i=1}^k B(x_i,r)\ne \emptyset. 
$$
Is it true that the convex hull of $X$ is contained in
$$
\bigcup_{i=1}^k B(x_i,r)   ?
$$
I needed a version of this in some Riemannian geometry argument and while I found a way around, the proof would have been a bit cleaner if this statement were true.

REPLY [11 votes]: Yes. Any point $y$ in the convex hull of $x$'s is a barycenter of some non-negative masses $m_i$ in $x_i$, $\sum m_i=1$, $y=\sum m_i x_i$. Point $y$ minimizes the moment of inertia $I(p)=\sum m_i |p-x_i|^2$, $p\in E^n$, just because $I(p)=I(y)+|p-y|^2$. In particular, it can not happen that all distances $|y-x_i|$ are at least $r$, but for some other point $p$ all distances $|y-x_i|$ are less than $r$.