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Extreme inequalities of general $L_p$ $μ$-projection body and general $L_p$ $μ$-centroid body

Chao Li, Gangyi Chen

Abstract

In this paper, we introduce the concept of general $L_p$ projection body and general $L_p$ centroid body of general measures with positive homogeneity density function, and prove the corresponding extreme inequalities. Meanwhile, we also study their measure comparison problem and monotone inequalities.

Extreme inequalities of general $L_p$ $μ$-projection body and general $L_p$ $μ$-centroid body

Abstract

In this paper, we introduce the concept of general projection body and general centroid body of general measures with positive homogeneity density function, and prove the corresponding extreme inequalities. Meanwhile, we also study their measure comparison problem and monotone inequalities.
Paper Structure (8 sections, 26 theorems, 173 equations)

This paper contains 8 sections, 26 theorems, 173 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. $L_p$-Minkowski combination and $L_p$-radial combination
  4. $L_p$ mixed $\mu$-measure
  5. $L_p$ dual mixed $\mu$ measure
  6. General $L_p$ $\mu$-projection bodies and General $L_p$ $\mu$-centroid bodies
  7. Extremal Inequalities for general $L_p$ $\mu$-projection body and $\mu$-centroid body
  8. Brunn-Minkowski Type Inequalities and Monotone inequalities

Key Result

Theorem 1.1

Let $\mu$ be a $1/s$ homogeneous Borel measure on $\mathbb{R}^n$ with $1/r$ homogeneous density function $\omega$, and $\omega$ is even, where $r>0$ and $s$ satisfy the equation $s=\frac{1}{n+\frac{1}{r}}$, if $K\in \mathscr{K}^n_o$, $p\geq 1$, and $\tau \in [-1,1]$, then when $\tau \neq 0$, equality holds in the left inequality if and only if $\Pi_{\mu,p}^{\tau}K$ is origin-symmetric, and when $

Theorems & Definitions (46)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Definition 2.2: LCC
  • Lemma 2.3: LCC
  • Lemma 2.4: LCC
  • ...and 36 more