SCHWEITZER 1985 Problem 2

Let $S$ be a given finite set of hyperplanes in $\mathbb{R}^n$ , and let $O$ be a point. Show that there exists a compact set $K \subseteq \mathbb{R}^n$ containing $O$ such that the orthogonal projection of any point of $K$ onto any hyperplane in $S$ is also in $K$ . (G.37) [Gy. Pap]