Minimal unfolded regions of a convex hull and parallel bodies

Abstract

The minimal unfolded region (or the heart) of a bounded subset Ω in the Euclidean space is a subset of the convex hull of Ω the definition of which is based on reflections in hyperplanes. It was introduced to restrict the location of the points that give extreme values of certain functions, such as potentials whose kernels are monotone functions of the distance, and solutions of differential equations to which Aleksandrov's reflection principle can be applied. We show that the minimal unfolded regions of the convex hull and parallel bodies of Ω are both included in that of Ω.