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Characterizations of Extremals for some Functionals on Convex Bodies

Published online by Cambridge University Press:  20 November 2018

Christos Saroglou
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Abstract

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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).

Keywords

52A4052A22
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 62 , Issue 6 , 14 December 2010 , pp. 1404 - 1418
Copyright
Copyright © Canadian Mathematical Society 2010

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