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

TL;DR

This paper investigates how certain affine surface areas behave under Steiner symmetrization, establishing monotonicity properties and deriving related affine isoperimetric inequalities without centroid assumptions.

Contribution

It proves monotonicity of $L_{ ext{phi}}$ and $L_{ ext{psi}}$ affine surface areas under Steiner symmetrization, extending affine isoperimetric inequalities without centroid constraints.

Findings

$L_{ ext{phi}}$ affine surface area is monotone increasing under Steiner symmetrization.

$L_{ ext{psi}}$ affine surface area is monotone decreasing under Steiner symmetrization.

Derived new affine isoperimetric inequalities under less restrictive conditions.

Abstract

In this paper, we prove that, if functions (concave) $\phi$ and (convex) $\psi$ satisfy certain conditions, the $L_{\phi}$ 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 $\phi$ and $\psi$, without assuming that the convex body involved has centroid (or the Santal\'{o} point) at the origin.