The heart of a convex body

We investigate some basic properties of the heart ♡(K) of a convex body K . It is a subset of K , whose definition is based on mirror reflections of Euclidean space, and is a non-local object. The main motivation of our interest for ♡(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 ♡(K) and the mirror symmetries of K ; we show that ♡(K) contains many (geometrically and physically) relevant points of K ; we prove a simple geometrical lower estimate for the diameter of ♡(K) ; we also prove an upper estimate for the area of ♡(K) , when K is a triangle.