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

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.