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An alternative definition for c-convex functions and another synthetic statement of MTW condition

Seonghyeon Jeong

TL;DR

The paper addresses how a fundamental regularity condition in optimal transport, the MTW condition, can be understood through a synthetic lens by introducing alternative c-convexity. It defines the c-chord and the notion of alternative c-convex functions, then proves a main theorem: c-convexity and alternative c-convexity are equivalent if and only if Loeper's property holds (equivalently MTW under adequate regularity). The work shows that, with Loeper, the two definitions align, while in its absence one can construct alt c-convex functions that fail to be c-convex, establishing a sharp boundary for the equivalence. This provides a purely geometric/synthetic characterization of MTW-type regularity conditions and connects them to simple convex-analytic ideas without requiring higher-order derivatives.

Abstract

The main theorem of this paper states that the c-convexity and the alternative c-convexity are equivalent if and only if the cost function c satisfies MTW condition. The alternative c-convex function is an analogy of the definition of the convex function that is using the inequality phi(t x1 + (1 - t) x0) <= t phi(x1) + (1 - t) phi(x0). We study properties of the alternative c-convex functions and MTW condition, then prove the main theorem.

An alternative definition for c-convex functions and another synthetic statement of MTW condition

TL;DR

The paper addresses how a fundamental regularity condition in optimal transport, the MTW condition, can be understood through a synthetic lens by introducing alternative c-convexity. It defines the c-chord and the notion of alternative c-convex functions, then proves a main theorem: c-convexity and alternative c-convexity are equivalent if and only if Loeper's property holds (equivalently MTW under adequate regularity). The work shows that, with Loeper, the two definitions align, while in its absence one can construct alt c-convex functions that fail to be c-convex, establishing a sharp boundary for the equivalence. This provides a purely geometric/synthetic characterization of MTW-type regularity conditions and connects them to simple convex-analytic ideas without requiring higher-order derivatives.

Abstract

The main theorem of this paper states that the c-convexity and the alternative c-convexity are equivalent if and only if the cost function c satisfies MTW condition. The alternative c-convex function is an analogy of the definition of the convex function that is using the inequality phi(t x1 + (1 - t) x0) <= t phi(x1) + (1 - t) phi(x0). We study properties of the alternative c-convex functions and MTW condition, then prove the main theorem.
Paper Structure (13 sections, 27 theorems, 374 equations)

This paper contains 13 sections, 27 theorems, 374 equations.

Table of Contents

  1. Introduction
  2. Conditions on the cost function
  3. Conditions on the cost function
  4. $c$-affine functions
  5. $c$-convex functions
  6. c-chord
  7. $c$-chord and basic properties (without Loeper's property)
  8. Properties of $c$-chord using Loeper's property
  9. Alternative c-convex functions
  10. Alternative $c$-convex functions
  11. Proof of the main theorem
  12. The first part of the main theorem
  13. Second part of the main theorem

Key Result

Theorem 1.1

Let $\mathbb{X}$ and $\mathbb{Y}$ be compact subsets of $\mathbb{R}^n$ with non-empty interior, and let $c: \mathbb{X} \times \mathbb{Y} \to \mathbb{R}$ be a $C^2$ function that satisfies bi-twist, non-degenerate. Also, assume that $\mathbb{X}$ and $\mathbb{Y}$ are $c$-convex with respect to each ot

Theorems & Definitions (75)

  • Theorem 1.1: Main theorem
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • ...and 65 more