The Heart of a Convex Body

Abstract

We investigate some basic properties of the heart $\heartsuit(\mathcal{K})$ 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 $\heartsuit(\mathcal{K})$ 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 $\heartsuit(\mathcal{K})$ and the mirror symmetries of $\mathcal{K}$; we show that $\heartsuit(\mathcal{K})$ contains many (geometrically and physically) relevant points of $\mathcal{K}$; we prove a simple geometrical lower estimate for the diameter of $\heartsuit(\mathcal{K})$; we also prove an upper estimate for the area of $\heartsuit(\mathcal{K})$, when $\mathcal{K}$ is a triangle.