$L_p$ geominimal surface areas and their inequalities

Deping Ye Memorial University of Newfoundland

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
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  title={$L_p$ geominimal surface areas and their inequalities},
  author={Deping Ye},
  booktitle={Int. Math. Res. Notes},
  pages={2465- 2498},
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.
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