This is an announcement for the paper "On the $L_p$ geominimal surface area and related inequalities" by Deping Ye.

Abstract: 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 the Santal'{o} style inequality. Some cyclic inequalities are established to obtain the monotonicity of the $L_p$ Geominimal surface area. Comparison between the $L_p$ Geominimal surface area and the (formal) $p$-surface area is also provided.

Archive classification: math.MG math.DG math.FA

Mathematics Subject Classification: 52A20, 53A15

Submitted from: deping.ye@gmail.com

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1308.4196

or

http://arXiv.org/abs/1308.4196