Some inequalities of isoperimetric type for the c-affine surface area

Abstract

We study the c-affine surface area ${\Omega }^c$, recently introduced by Schütt, Werner and Yalikun. We show that on the class of ball-bodies, ${\Omega }^c$ is maximized by a ball of radius $\frac{n}{n+1}$, and that a Santaló-type inequality holds: ${\Omega }^c(K){\Omega }^c(K^c)\le {\Omega }^c(\frac{1}{2}{B}_2^n{)}^2$. We also produce some more intricate inequalities involving the surface area.

9 pages