location:  Publications → journals → CJM
Abstract view

# Characterizations of Extremals for some Functionals on Convex Bodies

Published:2010-07-06
Printed: Dec 2010
• Christos Saroglou,
University of Crete, Department of Mathematics
 Format: HTML LaTeX MathJax PDF

## Abstract

We investigate equality cases in inequalities for Sylvester-type functionals. Namely, it was proven by Campi, Colesanti, and Gronchi that the quantity $$\int_{x_0\in K}\cdots\int_{x_n\in K}[V(\textrm{conv}\{x_0,\dots,x_n\})]^pdx_0\cdots dx_n , n\geq d, p\geq 1$$ 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 $h$: $\mathbb{R}_+\rightarrow \mathbb{R}_+$ is a strictly increasing function and $W_j$ is the $j$-th quermassintegral in $\mathbb{R}^d$, we prove that the functional $$\int_{x_0\in K_0}\cdots\int_{x_n\in K_n}h(W_j(\textrm{conv}\{x_0,\dots,x_n\}))dx_0\cdots dx_n , n \geq d$$ is minimized among the $(n+1)$-tuples of convex bodies of fixed volumes if and only if $K_0,\dots,K_n$ 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).
 MSC Classifications: 52A40 - Inequalities and extremum problems 52A22 - Random convex sets and integral geometry [See also 53C65, 60D05]

 top of page | contact us | privacy | site map |