The Heart of a Convex Body
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 mot
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Abstract We investigate some basic properties of the heart ♥(K) of a convex set 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. Keywords Convex bodies · Hot spots · Critical points · Shape optimization
1 Introduction Let K be a convex body in the Euclidean space RN , that is K is a compact convex set with non-empty interior. In [1] we defined the heart ♥(K) of K as follows. Fix a unit vector ω ∈ SN −1 and a real number λ; for each point x ∈ RN , let Tλ,ω (x) denote the reflection of x in the hyperplane πλ,ω of equation x, ω = λ (here, x, ω denotes the usual scalar product of vectors in RN ); then set Kλ,ω = x ∈ K : x, ω ≥ λ (see Fig. 1). The heart of K is thus defined as K−λ,−ω : Tλ,ω (Kλ,ω ) ⊂ K . ♥(K) = ω∈SN−1
L. Brasco (B) Laboratoire d’Analyse, Topologie, Probabilités, Aix-Marseille Université, CMI 39, Rue Frédéric Joliot Curie, 13453 Marseille Cedex 13, France e-mail: [email protected] R. Magnanini Dipartimento di Matematica “U. Dini”, Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy e-mail: [email protected] R. Magnanini et al. (eds.), Geometric Properties for Parabolic and Elliptic PDE’s, Springer INdAM Series 2, DOI 10.1007/978-88-470-2841-8_4, © Springer-Verlag Italia 2013
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L. Brasco and R. Magnanini
Fig. 1 The sets Kλ,ω and K−λ,−ω
Our interest in ♥(K) was motivated in [1] in connection to the problem of locating the (unique) point of maximal temperature—the hot spot—in a convex heat conductor with boundary temperature grounded at zero. There, by means of A.D. Aleksandrov’s reflection principle, we showed that ♥(K) must contain the hot spot at each time and must also contain the maximum point of the first Dirichlet eigenfunction of the Laplacian, which is known to control the asymptotic behavior of temperature for large times. By the same arguments, we showed in [1] that ♥(K) must also contain the maximum point of positive solutions of nonlinear equations belonging to a quite large class. By these observations, the set ♥(K) can be viewed as a geometrical means to estimate the positions of these important points. Another interesting feature of ♥(K) is the non-local nature of its definition. We hope that the study of ♥(K) can help, in a relatively simple setting, to develop techniques that may be useful in the study of other objects and properties of nonlocal nature, which have lately raised interest in the study of partial differential equations. A further reason of interest is that the shape of ♥(K)
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