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$\operatorname{SL}(n)$ contravariant function-valued valuations on polytopes

Zhongwen Tang, Jin Li, Gangsong Leng

Abstract

We present a complete classification of $\operatorname{SL}(n)$ contravariant, $C(\mathbb{R}^n\setminus\{o\})$-valued valuations on polytopes, without any additional assumptions.It extends the previous results of the second author [Int. Math. Res. Not. 2020] which have a good connection with the $L_p$ and Orlicz Brunn-Minkowski theory. Additionally, our results deduce a complete classification of $\operatorname{SL}(n)$ contravariant symmetric-tensor-valued valuations on polytopes.

$\operatorname{SL}(n)$ contravariant function-valued valuations on polytopes

Abstract

We present a complete classification of contravariant, -valued valuations on polytopes, without any additional assumptions.It extends the previous results of the second author [Int. Math. Res. Not. 2020] which have a good connection with the and Orlicz Brunn-Minkowski theory. Additionally, our results deduce a complete classification of contravariant symmetric-tensor-valued valuations on polytopes.
Paper Structure (5 sections, 16 theorems, 68 equations)

This paper contains 5 sections, 16 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. The new valuation $\Pi_{\zeta}$.
  4. Proof of main results
  5. Proof of the corollaries

Key Result

Theorem 1

Let $n\geq 3$. A map $Z:\mathcal{P}_{o}^n \rightarrow C(\mathbb{R}^n \setminus \{ o \})$ is an ${\operatorname{SL}(n)}$ contravariant valuation if and only if there are constants $c_{n-1},c_0,c_0'\in \mathbb{R}$ and a binary function $\zeta:\mathbb{R}\times (0,\infty)\rightarrow \mathbb{R}$ satisfy for every $P\in\mathcal{P}_o^n$ and $x\in \mathbb{R}^n \setminus \{o\}$.

Theorems & Definitions (26)

  • Theorem 1
  • Corollary 1
  • Corollary 2: Li li2020
  • Corollary 3
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 16 more