# MathSciDoc: An Archive for Mathematician ∫

#### Convex and Discrete Geometry mathscidoc:1806.40001

Indiana Univ. Math. J., 63, (1), 1-19, 2014
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.
affine surface area, $L_p$ Brunn-Minkowski theory, affine isoperimetric inequality, Steiner symmetrization, the Orlicz-Brunn-Minkowski theory
@inproceedings{deping2014on,
title={On the monotone properties of general affine surface areas under the Steiner symmetrization},
author={Deping Ye},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180610090159043400099},
booktitle={Indiana Univ. Math. J.},
volume={63},
number={1},
pages={1-19},
year={2014},
}

Deping Ye. On the monotone properties of general affine surface areas under the Steiner symmetrization. 2014. Vol. 63. In Indiana Univ. Math. J.. pp.1-19. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180610090159043400099.