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. 



covering convex sets by round balls
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 $|p-x\_i|$ are less than $r$.


