Some inequalities for (p,q)-mixed volume

Bin Chen,Weidong Wang

doi:10.1186/s13660-018-1836-2

Abstract

Lutwak, Yang, and Zhang introduced the concept of (p,q)-mixed volume whose special cases contain the L_{p}-mixed volume and the L_{p}-dual mixed volume. In this article, associated with the (p,q)-mixed volumes, we establish related cyclic inequalities, monotonic inequalities, and product inequalities.

Highlights

1 Introduction and main results At the end of the nineteenth century, Brunn and Minkowski pioneered the classical Brunn–Minkowski theory of convex bodies, which is the product of Minkowski linear combination of vectors and volumes in the Euclidean space
The classical dual Brunn–Minkowski theory of star bodies was introduced by Lutwak [24] in 1975
Proof of Theorem 1.6 For p, q ∈ R and q > n, K, L, M ∈ Kon. From the definitions of support function and radial function, we know ρM(u) ≤ hM(u) with equality if and only if M is a ball centered at the origin

Summary

Introduction

Introduction and main resultsAt the end of the nineteenth century, Brunn and Minkowski pioneered the classical Brunn–Minkowski theory of convex bodies, which is the product of Minkowski linear combination of vectors and volumes in the Euclidean space. The classical dual Brunn–Minkowski theory of star bodies was introduced by Lutwak [24] in 1975. In 1996, on the basis of Lp harmonic radial combination, Lutwak [23] put forward the concept of Lp-dual mixed volume (p ≥ 1) and gave its integral expression.

Results

Conclusion