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

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