Calculus of Variations and Geometric Measure Theory

L. Brasco - R. Magnanini

The heart of a convex body

created by brasco on 23 Feb 2012
modified on 27 Aug 2014


Accepted Paper

Inserted: 23 feb 2012
Last Updated: 27 aug 2014

Journal: Springer INdAM Series
Pages: 15
Year: 2012

This paper has been written for possible publication in the proceedings of the conference "Geometric properties for parabolic and elliptic PDEs", held in Cortona in June 2011


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 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.

Keywords: Hot spot, eigenfunctions, convex bodies, Fraenkel asymmetry