# MathSciDoc: An Archive for Mathematician ∫

#### Convex and Discrete Geometry mathscidoc:1806.40002

Int. Math. Res. Notes, 2015, 2465- 2498, 2015
In this paper, we introduce the \$L_p\$ geominimal surface area for all \$-n\neq p<1\$, which extends the classical geominimal surface area (\$p=1\$) by Petty and the \$L_p\$ geominimal surface area by Lutwak (\$p>1\$). Our extension of the \$L_p\$ geominimal surface area is motivated by recent work on the extension of the \$L_p\$ affine surface area -- a fundamental object in (affine) convex geometry. We prove some properties for the \$L_p\$ geominimal surface area and its related inequalities, such as, the affine isoperimetric inequality and a Santal\'{o} style inequality. Cyclic inequalities are established to obtain the monotonicity of the \$L_p\$ geominimal surface areas. Comparison between the \$L_p\$ geominimal surface area and the \$p\$-surface area is also provided.
affine surface area, \$L_p\$ affine surface area, geominimal surface area, \$L_p\$ geominimal surface area, \$L_p\$-Brunn-Minkowski theory, affine isoperimetric inequalities, the Blaschke-Santal\'{o} inequality, the Bourgain-Milman inverse Santal\'o inequality
```@inproceedings{deping2015\$l_p\$,
title={\$L_p\$ geominimal surface areas and their inequalities},
author={Deping Ye},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180610092158200185100},
booktitle={Int. Math. Res. Notes},
volume={2015},
pages={2465- 2498},
year={2015},
}
```
Deping Ye. \$L_p\$ geominimal surface areas and their inequalities. 2015. Vol. 2015. In Int. Math. Res. Notes. pp.2465- 2498. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180610092158200185100.