Let $H_1,H_2,\dots,H_n$ be compact and convex sets in $\mathbb{R}^n$ such that $\bigcap_{j=1}^n H_j$ has non-empty interior and for each $i=1,2,\dots,n$ there exist at least one element $x \in H_i$ such that $x \not \in \bigcup_{j\not = i} H_j$. Can $\bigcap_{j=1}^n \partial H_j$ be a single point?