Characterizations of Extremals for some Functionals on Convex Bodies

Canadian journal of mathematics, Tome 62 (2010) no. 6, pp. 1404-1418

Résumé

We investigate equality cases in inequalities for Sylvester-type functionals. Namely, it was proven by Campi, Colesanti, and Gronchi that the quantity $$\int_{{{x}_{0}}\in K}{\cdot \cdot \cdot \int_{{{x}_{n}}\in K}{{{\left[ V\left( \text{conv}\left\{ {{x}_{0}},\cdot \cdot \cdot ,{{x}_{n}} \right\} \right) \right]}^{p}}d{{x}_{0}}\cdot \cdot \cdot }\,d{{x}_{n}},n\ge d,p\ge 1}$$ is maximized by triangles among all planar convex bodies $K$ (parallelograms in the symmetric case). We show that these are the only maximizers, a fact proven by Giannopoulos for $p\,=\,1$ . Moreover, if $h:\,{{R}_{+}}\,\to \,{{R}_{+}}$ is a strictly increasing function and [{{W}_{j}}$ is the $j$ -th quermassintegral in ${{R}^{d}}$ , we prove that the functional $$\int_{{{x}_{0}}\in {{K}_{0}}}{\cdot \cdot \cdot }\int_{{{x}_{n}}\in {{K}_{n}}}{h\left( {{W}_{j}}\left( \text{conv}\left\{ {{x}_{0}},\cdot \cdot \cdot ,{{x}_{n}} \right\} \right) \right)}\,d{{x}_{0}}\cdot \cdot \cdot d{{x}_{n}},n\ge d$$ is minimized among the $(n\,+\,1)$ -tuples of convex bodies of fixed volumes if and only if ${{K}_{0,\,\ldots \,,}}{{K}_{n}}$ are homothetic ellipsoids when $j\,=\,0$ (extending a result of Groemer) and Euclidean balls with the same center when $j\,>\,0$ (extending a result of Hartzoulaki and Paouris).

Détail

DOI : 10.4153/CJM-2010-062-2

Mots-clés : 52A40, 52A22