The heart of a convex body

Abstract

We investigate some basic properties of the {\it 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 phisically) 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.