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

TL;DR

This paper investigates inequalities related to the c-affine surface area, establishing maximization and Santaló-type inequalities for specific convex bodies and exploring additional surface area inequalities.

Contribution

It introduces new inequalities for the c-affine surface area, including maximization on ball-bodies and a Santaló-type inequality, advancing understanding in convex geometry.

Findings

Maximization of c-affine surface area by specific ball-bodies.

A Santaló-type inequality for c-affine surface area.

Additional inequalities involving surface area.

Abstract

We study the c-affine surface area $\Omega^c$, recently introduced by Sch\"utt, 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 Santal\'o-type inequality holds: $\Omega^c(K) \Omega^c(K^c) \leq \Omega^c(\frac{1}{2} B_2^n)^2$. We also produce some more intricate inequalities involving the surface area.