Boundedly Polyhedral Sets and F-Simplices

Generalizing the concept of Choquet simplex, we study a new class of convex solids $K$ in $\mathbb{R}^n$ which satisfy the following condition: all $n$-dimensional intersections of the form $K \cap (x + K)$, $x \in \mathbb{R}^n$, belong to at most finitely many homothety classes of convex solids. Our description of this class uses new results on boundedly polyhedral sets.