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Homogeneous optimal transport maps between oblique cones

Tristan C. Collins, Benjy Firester, Freid Tong

TL;DR

The paper addresses the existence and boundary regularity of homogeneous optimal transport maps between convex cones with homogeneous densities under a strong obliqueness condition. It develops a variational framework that reduces the conic OT problem to a free boundary Monge–Ampère equation on the links of the cones, and proves the existence of homogeneous OT maps with $C^{1,\varepsilon}$ regularity under strong obliqueness. It further extends the analysis to strongly partially oblique cones by introducing a normalization to handle translational invariance and obtaining analogous existence and regularity results. The results connect to tangent-cone analyses relevant for boundary regularity of OT maps and have implications for constructing complete Calabi–Yau metrics on certain quasi-projective varieties.

Abstract

We construct homogeneous optimal transport maps for the quadratic cost between convex cones with homogeneous, possibly degenerate, densities when the cones satisfy an obliqueness condition. The existence of such maps plays a central role in the boundary regularity theory for optimal transport maps between convex domains. Our results are also relevant for the existence of complete Calabi-Yau metrics on certain quasi-projective varieties.

Homogeneous optimal transport maps between oblique cones

TL;DR

The paper addresses the existence and boundary regularity of homogeneous optimal transport maps between convex cones with homogeneous densities under a strong obliqueness condition. It develops a variational framework that reduces the conic OT problem to a free boundary Monge–Ampère equation on the links of the cones, and proves the existence of homogeneous OT maps with regularity under strong obliqueness. It further extends the analysis to strongly partially oblique cones by introducing a normalization to handle translational invariance and obtaining analogous existence and regularity results. The results connect to tangent-cone analyses relevant for boundary regularity of OT maps and have implications for constructing complete Calabi–Yau metrics on certain quasi-projective varieties.

Abstract

We construct homogeneous optimal transport maps for the quadratic cost between convex cones with homogeneous, possibly degenerate, densities when the cones satisfy an obliqueness condition. The existence of such maps plays a central role in the boundary regularity theory for optimal transport maps between convex domains. Our results are also relevant for the existence of complete Calabi-Yau metrics on certain quasi-projective varieties.

Paper Structure

This paper contains 4 sections, 19 theorems, 103 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and variational framework
  3. Minimizing solutions for strongly oblique cones
  4. Strongly partially oblique cones

Key Result

Theorem 1.1

Suppose two strict cones $(\mathtt{C}, d\mu)$ and $(\mathtt{C}',d\nu)$ are strongly oblique with homogeneous, locally finite, doubling densities. Then, there exists a homogeneous optimal transport map $\nabla \varphi$ between them for $\varphi \in C^{1,\epsilon}(\overline{\mathtt{C}(P)})$ for some $

Figures (1)

  • Figure 1: The pair $(P,\Sigma)$ is drawn in green and blue respectively, which is strongly oblique as seen by $-P$ in red being contained in $\Sigma^\circ$ in yellow

Theorems & Definitions (44)

  • Definition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Definition 1.2
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Remark 2.1
  • Lemma 2.2
  • ...and 34 more