General L p $L_{p}$ -mixed chord integrals of star bodies

Abstract

The notion of general mixed chord integrals of star bodies was introduced by Feng and Wang. In this paper, we extend the concept of the general mixed chord integrals to general L p -mixed chord integrals of star bodies. Based on this new notion, we study their extremum values and obtain an Aleksandrov-Frenchel type and a cyclic inequality for general L p -mixed chord integrals of star bodies, respectively. Further, as applications, we establish two Brunn-Minkowski type inequalities for L p -radial bodies. Finally, we get an interesting identical equality on combining L p -radial bodies.MSC: 52A20; 52A40; 52A39

Cyclic Brunn–Minkowski inequalities for general width and chord-integrals

“…Our works belong to the asymmetric Brunn-Minkowski theory, which has its starting point in the theory of valuations in connection with isoperimetric and analytic inequalities. As an important research object in convex geometry, asymmetric Brunn-Minkowski theory has gotten rich development, readers can refer to [12][13][14][15][16][17][18][19].…”

General Blaschke Bodies and the Asymmetric Negative Solutions of Shephard Problem

“…If M is centrally symmetric, i.e., M = −M + m, m ∈ R n , then from (5), (17), (21), and (22), we have…”

Inequalities on General Lp-Mixed Chord Integral Difference

“…In 2016, Li and Wang extended the general mixed chord-integral to the general L pmixed chord integral of star bodies (see [15]…”

“…said to have a similar general L p -chord. For this general L p -chord integral, Li and Wang gave the following inequalities (see [15]).…”