The heart of a convex body

TL;DR

This paper studies the properties of the 'heart' of a convex body, a non-local set related to mirror symmetries, which helps estimate the hot spot location in convex heat conductors.

Contribution

It establishes geometric bounds on the diameter and area of the heart, and explores its relation to symmetries and physical heat distribution models.

Findings

The heart contains key points related to the body's symmetries.

A lower bound for the diameter of the heart is provided.

An upper bound for the area of the heart in triangles is derived.

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.