L p $L_{p}$ -Dual geominimal surface area and general L p $L_{p}$ -centroid bodies

Jinsheng Guo,Yibin Feng

doi:10.1186/s13660-015-0888-9

Abstract

In this article, we consider the Shephard type problems and obtain the affirmative and negative parts of the version of $L_{p}$ -dual geominimal surface area for general $L_{p}$ -centroid bodies. Combining with the $L_{p}$ -dual geominimal surface area we also give a negative form of the Shephard type problems for $L_{p}$ -centroid bodies.

Highlights

1 Introduction and main results Let Kn denote the set of convex bodies in Euclidean space Rn
For the set of convex bodies containing the origin in their interiors and the set of origin-symmetric convex bodies in Rn, we write Kon and Kcn, respectively
The notion of geominimal surface area was discovered by Petty

Summary

Introduction

Introduction and main results LetKn denote the set of convex bodies (compact, convex subsets with nonempty interiors) in Euclidean space Rn. For K ∈ Kn, the geominimal surface area, G(K), of K is defined by ωnn G(K ) = inf nV (K , Q)V Based on Lp-mixed volume, Lutwak [ ] introduced the notion of Lp-geominimal surface area. For K ∈ Kon, p ≥ , the Lp-geominimal surface area, Gp(K), of K is defined by p p ωnn Gp(K ) = inf nVp(K , Q)V Q∗ n : Q ∈ Kon .

Results

Conclusion