Characterizations of Extremals for some Functionals on Convex Bodies

http://dx.doi.org/10.4153/CJM-2010-062-2
Canad. J. Math. 62(2010), 1404-1418

  Published:2010-07-06
 Printed: Dec 2010

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).