*Accepted Paper*

**Inserted:** 23 feb 2012

**Last Updated:** 27 aug 2014

**Journal:** Springer INdAM Series

**Pages:** 15

**Year:** 2012

**Notes:**

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

**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 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

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