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arxiv: 2311.15655 · v4 · pith:JKICM7KHnew · submitted 2023-11-27 · 🧮 math.AP

Optimal (partial) transport to non-convex polygonal domains

Shibing Chen , Yuanyuan Li , Jiakun Liu This is my paper

Pith reviewed 2026-05-24 06:24 UTC · model grok-4.3

classification 🧮 math.AP
keywords optimal transportpartial transportsingular setfree boundarynon-convex polygonal domainregularitytwo dimensions
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The pith

Optimal transport to non-convex polygonal domains in the plane has a singular set that is a smooth curve away from finitely many points.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper investigates optimal transport problems with a non-convex polygonal target in two dimensions. It establishes that the singular set of the optimal map is locally a smooth one-dimensional curve except at finitely many points. The same regularity holds for the free boundary in the partial transport problem. A sympathetic reader would care because these statements extend regularity theory to targets with corners and indentations that appear in many applications, while the higher-dimensional conjectures point toward a broader geometric picture for polytope targets.

Core claim

For the complete optimal transport problem, we prove that the singular set is locally a smooth one-dimensional curve away from finitely many points. For the optimal partial transport problem, we prove that the free boundary is smooth away from finitely many singular points. In higher dimensions, we formulate two conjectures concerning the structure of singularities when the target is a non-convex polytope.

What carries the argument

The flat sides and finite vertices of the non-convex polygonal target in R^2, which are used to localize and control the geometry of the singular set or free boundary.

If this is right

  • The optimal map is smooth except along a one-dimensional curve that meets the boundary or itself only at finitely many points.
  • The free boundary in the partial-transport setting obeys the same local smoothness description.
  • These local descriptions continue to hold when the measures and cost satisfy the usual existence hypotheses.
  • Analogous but possibly more intricate singularity structures are conjectured to exist for non-convex polytope targets in dimensions greater than two.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The two-dimensional results may serve as a model for approximating domains with piecewise-smooth boundaries by polygons.
  • Numerical methods could explicitly track the one-dimensional singular curves rather than treating them as fully unstructured.
  • Techniques developed here might transfer to related free-boundary problems whose targets have flat faces.

Load-bearing premise

The target domain is a non-convex polygon in two dimensions, with the source measure, target measure, and cost satisfying the standard conditions that guarantee existence of an optimal map.

What would settle it

An explicit pair of measures on a non-convex quadrilateral target whose optimal map has a singular set containing a point that is not locally a smooth curve or that accumulates infinitely many exceptional points.

Figures

Figures reproduced from arXiv: 2311.15655 by Jiakun Liu, Shibing Chen, Yuanyuan Li.

Figure 3.1
Figure 3.1. Figure 3.1: Proposition 3.1. For any x ∈ Σ ′ 2 , there exists a small constant rx such that Brx (x) ∩ A is a smooth curve. Proof. Let x0 ∈ Σ ′ 2 . By a translation of coordinates, we may assume x0 = 0. By the definition of Σ′ 2 , ∂ −u(0) is a segment with endpoints on two open edges of Ω∗ , denoted by (bi1 bi1+1) and (bi2 bi2+1). Let y0 = ∂ −u(0) ∩ (bi1 bi1+1) and ˆy0 = ∂ −u(0) ∩ (bi2 bi2+1); see [PITH_FULL_IMAGE:f… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Proof. Let x0 ∈ Σ1 ∪ Σ ′′ 2 . We may assume that Bδ(x0) ∩ A \ {x0} ̸= ∅ for any δ > 0, as otherwise x0 would be an isolated singularity, and the proof is done. By Lemma 3.2, we have that ∂ −u(x0)∩∂Ω ∗ can be decomposed into a finite number of connected components Ci , i = 1, . . . , l, each being either a point or an edge. By a translation of coordinates, we may assume x0 = 0. By subtracting a constant, … view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Lemma 4.3. Both F1 and F2 consist of finitely many points. Proof. For any x ∈ F1, by the definition of F1, there exists a vertex bi of Ω∗ , 1 ≤ i ≤ m, such that Dv(bi) = x. This implies that F1 is a finite set with a cardinality no greater than m. Let x ∈ F2. By the definition of F2, there are two distinct integers i, j, 1 ≤ i ̸= j ≤ m, such that ∂ −u¯(x) ∩ (bibi+1) ̸= ∅ and ∂ −u¯(x) ∩ (bj bj+1) ̸= ∅. De… view at source ↗
read the original abstract

In this paper, we investigate optimal (partial) transport problems for which the target is a non-convex polygonal domain in \(\mathbb{R}^2\). For the complete optimal transport problem, we prove that the singular set is locally a smooth one-dimensional curve away from finitely many points. For the optimal partial transport problem, we prove that the free boundary is smooth away from finitely many singular points. In higher dimensions, we formulate two conjectures concerning the structure of singularities when the target is a non-convex polytope.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript investigates optimal (partial) transport problems where the target is a non-convex polygonal domain in R^2. For the complete optimal transport problem, it proves that the singular set is locally a smooth one-dimensional curve away from finitely many points. For the optimal partial transport problem, it proves that the free boundary is smooth away from finitely many singular points. Two conjectures are formulated for the structure of singularities in higher dimensions when the target is a non-convex polytope.

Significance. If the results hold, they advance regularity theory for optimal transport and partial transport with non-convex targets, providing explicit geometric descriptions (smooth 1D curves or surfaces away from finite points) that extend beyond convex-domain results. The 2D theorems are stated as direct proofs, and the higher-dimensional conjectures are clearly posed. These could support further work on singularity structure in geometric PDEs.

minor comments (1)
  1. The abstract is clear, but the introduction should explicitly list the standing assumptions on the source/target measures and cost function to make the well-posedness statements self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending acceptance. The report contains no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states direct mathematical proofs of regularity for the singular set (smooth 1D curve away from finitely many points) and free boundary (smooth away from finitely many singular points) in optimal (partial) transport to non-convex polygonal domains in R^2, under standard well-posedness conditions. No equations, parameters, or claims in the abstract reduce by construction to inputs, fitted data, or self-citation chains; the results are presented as theorems derived from the geometric setting rather than renamed or self-referential constructs. This is a standard pure-math regularity paper with no load-bearing self-citations or ansatz smuggling visible.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information available from the abstract to identify specific free parameters, axioms, or invented entities; the paper consists of regularity proofs rather than introducing new fitted quantities or entities.

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Works this paper leans on

31 extracted references · 31 canonical work pages

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