Let $H_1,H_2,\dots,H_n$ be compact and convex sets in $\mathbb{R}^n$ such that $\cap_{j=1}^n H_j$ has non-empty interior and $H_i \not = \cap_{j=1}^n H_j$ for all $i=1,2,\dots,n$. Can $\cap_{j=1}^n \partial H_j$ be a single point?