The heart of a convex body

Abstract

We investigate some basic properties of the {\it heart} $♡(\mathcal{K})$ of a convex set $\mathcal{K}.$ It is a subset of $\mathcal{K},$ whose definition is based on mirror reflections of euclidean space, and is a non-local object. The main motivation of our interest for $♡(\mathcal{K})$ is that this gives an estimate of the location of the hot spot in a convex heat conductor with boundary temperature grounded at zero. Here, we investigate on the relation between $♡(\mathcal{K})$ and the mirror symmetries of $\mathcal{K};$ we show that $♡(\mathcal{K})$ contains many (geometrically and phisically) relevant points of $\mathcal{K};$ we prove a simple geometrical lower estimate for the diameter of $♡(\mathcal{K});$ we also prove an upper estimate for the area of $♡(\mathcal{K}),$ when $\mathcal{K}$ is a triangle.

15 pages, 3 figures. appears as "Geometric Properties for Parabolic and Elliptic PDE's", Springer INdAM Series Volume 2, 2013, pp 49-66

References in corpus (1)

• Renormalization of potentials and generalized centers

Cited by in corpus (1)

• Minimal unfolded regions of a convex hull and parallel bodies