Minimal unfolded regions of a convex hull and parallel bodies

TL;DR

This paper investigates the properties of minimal unfolded regions (hearts) of convex hulls and parallel bodies, showing they are contained within the minimal unfolded region of the original set, with implications for potential theory and differential equations.

Contribution

It establishes that the minimal unfolded regions of convex hulls and parallel bodies are contained within that of the original set, extending understanding of geometric reflection properties.

Findings

Minimal unfolded regions of convex hulls are contained within those of the original set.

Parallel bodies' minimal unfolded regions are also included in the original set's region.

Results have implications for potential functions and solutions to differential equations.

Abstract

The {\em minimal unfolded region} (or the {\em heart}) of a bounded subset $\Om$ in the Euclidean space is a subset of the convex hull of $\Om$ 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. %the temperature of a heat conductor with some initial-boundary condition, in which case the points are called the hot spots. We show that the minimal unfolded regions of the convex hull and parallel bodies of $\Om$ are both included in that of $\Om$.