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A characterization of the Legendre involution on the class of generic frontals

Takashi Nishimura

Abstract

We show that, under an additional mild assumption, on the class of generic frontals, any involution whose fixed point set is exactly the same as the fixed point set of the Legendre involution must be the Legendre involution (Theorem 2 in §1). Moreover, its natural complexification (Corollary 1 in §3) is simultaneously shown.

A characterization of the Legendre involution on the class of generic frontals

Abstract

We show that, under an additional mild assumption, on the class of generic frontals, any involution whose fixed point set is exactly the same as the fixed point set of the Legendre involution must be the Legendre involution (Theorem 2 in §1). Moreover, its natural complexification (Corollary 1 in §3) is simultaneously shown.

Paper Structure

This paper contains 5 sections, 9 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Envelopes created by hyperplane families
  4. Natural correspondence $\mathcal{C}: \mathcal{F}\to \mathcal{X}$
  5. Proof of Theorem \ref{['theorem2']}

Key Result

Theorem 1

Assume a transform $\widetilde{\mathcal{T}} : Cvx \left(\mathbb{R}^n\right)\to Cvx \left(\mathbb{R}^n\right)$ satisfies Then, $\widetilde{\mathcal{T}}$ is essentially the classical Legendre transform $\widetilde{\mathcal{L}}: Cvx\left(\mathbb{R}^n\right)\to Cvx\left(\mathbb{R}^n\right)$; namely there exists a constant $C_0\in \mathbb{R}$, a vector $v_0\in \mathbb{R}^n$, and an invertible symmetri

Theorems & Definitions (16)

  • Theorem 1: artsteinavidanmilman
  • Definition 1
  • Definition 2: Appendix 4 in arnoldmechanics
  • Theorem 2
  • Definition 3: Definition 1 in nishimura
  • Definition 4: A strong version of Definition 2 in nishimura
  • Remark 2.1
  • Theorem 3: (a) of Thorem 1 in nishimura
  • Theorem 4: (b) of Theorem 1 in nishimura
  • Example 2.1
  • ...and 6 more