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Characterizations of Extremals for some Functionals on Convex Bodies


Published:20100706
Printed: Dec 2010
Christos Saroglou,
University of Crete, Department of Mathematics
Abstract
We investigate equality cases in inequalities for Sylvestertype
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).