On the monotone properties of general affine surface areas under the Steiner symmetrization

Abstract

In this paper, we prove that, if functions (concave) $\varphi$ and (convex) $\psi$ satisfy certain conditions, the $L_{\varphi }$ affine surface area is monotone increasing, while the $L_{\psi }$ affine surface area is monotone decreasing under the Steiner symmetrization. Consequently, we can prove related affine isoperimetric inequalities, under certain conditions on $\varphi$ and $\psi$, without assuming that the convex body involved has centroid (or the Santaló point) at the origin.

References in corpus (3)

• General affine surface areas
• Relative entropy of cone measures and $L_p$ centroid bodies
• A note on Mahler's conjecture

Cited by in corpus (4)

• The Dual Orlicz-Brunn-Minkowski Theory
• Dual Orlicz-Brunn-Minkowski theory: dual Orlicz $L_{\varphi }$ affine and geominimal surface areas
• New Orlicz Affine Isoperimetric Inequalities
• Extremal general affine surface areas