{'fraction_non_alphanumeric': 0.06578273123238534, 'fraction_numerical': 0.021201639764283884, 'mean_word_length': 3.792203806015961, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Prohibiting isolated singularities in optimal transport

We give natural topological conditions on the support of the target measure under which solutions to the optimal transport problem with cost function satisfying the (weak) Ma, Trudinger, and Wang condition cannot have any isolated singular points.


Main results
The optimal transport problem is the following: given source and target probability measure spaces (Ω, µ), Ω , ν , and a measurable cost function c : Ω ×Ω → R, find an optimal measurable mapping T : Ω →Ω defined µ-a.e., minimizing Ω c(x, F (x))µ(dx) (1) over the set of all measurable F : Ω →Ω with F # µ = ν. A fundamental problem in optimal transport theory is to understand the regularity of optimal maps. In the classical case where the cost function is given by the quadratic cost c(x,x) = |x −x| 2 /2 on R n ×R n (or equivalently c(x,x) = − x,x ), it is well known that the optimal map is Hölder continuous [5][6][7] if the support of the target measure is convex, for source and target measures with densities bounded above and below. For more general cost functions one would require certain structural conditions, namely, (Twist), (Nondeg) and especially, the c-convexity of the support of the target measure and the condition (MTW) [24,27] which are shown to be necessary [21,24] for regularity theory of the classical case to be extended (see Section 2 for relevant definitions). Under these conditions, Hölder continuity of the optimal map is known, under the assumption that the source and target measures have densities bounded from above and below, see [12,14,19,21]. For smoother measures, higher regularity theory is also known, see [20,24,27].
A natural question one can ask is what happens if one of the above structural conditions is violated. In particular, we focus on the geometric condition of convexity / c-convexity of the support of the target measure, where it is known that without such conditions optimal maps may not be continuous [7,24].
As a first step in this direction, in this paper we analyze the case of isolated singular points (thoughout this paper, by singularity or singular point we indicate a point where a function is not differentiable). In the following main theorem we prove that if the support of the target measure has no holes (by which we mean a bounded, open, connected component of the complement of the target domain), then the corresponding Brenier solution cannot have an isolated singular point in the interior of the support of the source measure. The precise statement follows with the relevant definitions given in Section 2.
Theorem 1.1. Let M andM be n-dimensional Riemannian manifolds and Ω andΩ be bounded domains in M andM , respectively. Let c be a cost function c : Ω ×Ω → R that satisfies (A0), (Twist), (Nondeg), and (MTW), and assume that Ω andΩ are c-convex with respect to each other.
Consider two absolutely continuous probability measures µ = f dVol and ν = g dVol on M andM , respectively, with supports spt µ ⊂ Ω and spt ν ⊂Ω. Assume that spt ν ∩Ω ∂ = ∅ and that there exists a constant 0 < Λ < ∞ such that on their supports. Finally, let u be a Brenier solution (see Definition 2.6) to the optimal transport problem with cost c. For each x 0 ∈ (spt µ) int , if there are no holes (see Definition 2.11) in spt ν that are c-convex with respect to x 0 , then x 0 cannot be an isolated singular point of u. Remark 1.2. Examples of cost functions satisfying (A0), (Twist), (Nondeg), and (MTW), can be found in [18,22,24,27], see also [28].
In the special case when M andM are domains in Euclidean space and c(x,x) = 1 2 |x −x| 2 , c-convexity reduces to ordinary convexity (we will henceforth refer to this setting as the Euclidean case). In the two dimensional Euclidean case, Figalli [9] studied the geometric structure of the singular set, and the above result on isolated singularity follows as a special case. In higher dimensions, it seems that no result on the geometric structure of singular sets (similar to the one in [9]) is currently known. For some previous related works in the Euclidean case, see [2,16], [29], and [8,Section 5] in the case of dimension 2, and [3,25] in higher dimensions. While the other results mentioned above consider isolated singularities of the Monge-Ampère equation, the papers [29] and [8,Section 5] deal specifically with the case of the optimal transport problem (however, still in the Euclidean case). Both results discuss the finer question of Lipschitz or C 1 propagation of singularities, but assume stronger conditions in addition to topological restrictions on the support of the target measure. Specifically, [29] assumes that all singular points have a subdifferential of affine dimension at most one, while [8] requires the support of the source measure be convex. Our main result applies to a more general class of c, and also requires no hypothesis on spt µ; in fact we obtain the condition required for [29] in the course of our proof (see Proposition 3.2).
Throughout this paper, we will denote the closure, interior, and boundary of a set A by A cl , A int , and A ∂ respectively.

Relevant definitions and preliminaries
In this section we gather some relevant definitions and facts about c-convex potential functions in relation to solutions of the optimal transport problem. Some good references are [15,28].
Let M andM be n-dimensional Riemannian manifolds and Ω andΩ be bounded domains in M andM , respectively. Let c be a measurable cost function c : Ω×Ω → R. We start out by stating the various assumptions we may require on our cost function c.

Smoothness of cost:
c ∈ C 4 (Ω cl ×Ω cl ). (A0) Twist: We will say c satisfies condition (Twist) if each of the mappings are injective for each x 0 ∈ Ω andx 0 ∈Ω. Here, D,D denote the usual differential in the x orx variable.
Remark 2.1. We use the standard notation exp c x0 (·) and exp c x0 (·) to denote the inverses of the above two mappings. Also, for any A ⊂ Ω,x ∈Ω orĀ ⊂Ω, x ∈ Ω, we will write . We also comment here, for the cost c(x,x) = − x,x on R n × R n , these mappings are both just the identity map.
Definition 2.2 (c-convexity of a set [24]). If A ⊂ Ω andx ∈Ω, we say that A is c-convex with respect tox if the set [A]x is a convex subset of T * xM . IfĀ ⊂Ω and x ∈ Ω, we define whenĀ is c-convex with respect to x and A andĀ are c-convex with respect to each other in the obvious way.

Nondegeneracy:
We say c satisfies condition (Nondeg) if, for each x ∈ Ω andx ∈Ω, the linear mapping is invertible (and consequently, so is its adjoint mapping, −DDc(x,x) : MTW (Ma-Trudinger-Wang condition) [21,24,27]: We say c satisfies the con- Here we fix coordinate systems on M andM and take all derivatives with respect to these coordinates; lower indices before a comma denote derivatives of c with respect to the x variable, and lower indices after a comma denote derivatives with respect to thex variable. Also, a pair of raised indices denotes the inverse of a matrix. We next define some basic concepts of use in c-convex geometry.
for all x ∈ Ω. Any function of the form −c(·,x 0 ) + λ 0 is called a c-affine function (with focusx 0 ), and if it satisfies the above relations is said to support u from below at x 0 .
We also define the c-subdifferential of a c-convex function, and the subdifferential of a semi-convex function.
Similarly, the c-subdifferential of a c-convex function u at a point x ∈ (dom (u)) int is defined as the set Remark 2.5. Note that if u is semi-convex, each ∂u (x) is a nonempty, convex set, and for any point Additionally, it is known that if c satisfies (A0), then a c-convex function is semiconvex, hence in particular it is differentiable a.e. Additionally, if u is c-convex it is not difficult to see that its c-subdifferential is c-monotone, i.e. for any Definition 2.6. Suppose c satisfies (Twist). A Brenier solution (to the optimal transport problem with cost c(x,x)) pushing µ forward to ν is a c-convex function u defined on spt µ such that ). If u is a Brenier solution pushing µ forward to ν, then it is well known that T as defined above is optimal in (1).
The following result (discovered by Loeper [21] for domains in R n , further developed in [13,17,23,26], and extended to domains in manifolds under certain conditions) details certain geometric properties of c-convex functions. It will play a key role in our main proof. Theorem 2.7 (Loeper's maximum principle [21]). Suppose c, Ω, andΩ satisfy the conditions of Theorem 1.1. Also let x 0 ∈ Ω,p 0 , An analogous inequality holds with the roles of Ω andΩ reversed.
This lemma has several important consequences, we will require the following two of them later; the second of which was first observed and used in [11,12] and [19]. . Suppose c, Ω, andΩ satisfy the same conditions as Theorem 2.7 above, and u is a c-convex function on Ω. Then for any in particular, ∂ c u(x 0 ) is c-convex with respect to x 0 .
Finally, we give the precise definition of a hole.

Proof of Theorem 1.1
We begin by deriving several intermediate results. We start with stating a very useful tool in our analysis, due to Albano and Cannarsa: The next result excludes having a full dimensional subdifferential at an isolated singular point, when the target domain contains no holes. Note that the result can be shown under just the condition (Twist), and can be strengthened under (Nondeg) and (MTW). We also comment that this will be the only place where we use the no-hole condition on spt ν, for the proof of Theorem 1.1.

Proposition 3.2.
Suppose that c is C 1 and satisfies (Twist), u is a c-convex Brenier solution, and spt ν contains no holes. Then u cannot have any isolated singular point x 0 ∈ (spt µ) int with affdim ∂u (x 0 ) = n (here affdim is the affine dimension of a convex set).
If in addition, c satisfies (A0), (Nondeg), and (MTW), and Ω andΩ are cconvex with respect to each other, we obtain the same conclusion under the weaker condition that spt ν contains no holes c-convex with respect to x 0 .
Proof. Suppose by contradiction that x 0 ∈ (spt µ) int is an isolated singular point of u, and the affine dimension of ∂u (x 0 ) is n. Since c is C 1 and satisfies (Twist), the mapping exp c x0 (·) is continuous and injective, thus Brouwer's invariance of domain theorem (see [4]) gives that exp c x0 (·) is a homeomorphism between the open set ∂u (x 0 ) int and its image. In particular, exp c x0 (∂u (x 0 ) int ) is a nonempty, open, bounded, connected set. Then since x 0 is an isolated singularity, by Proposition 3.1 we have as Du(dom (Du)) ⊂ spt ν for the Brenier solution u.
We now claim that for any x 1 = x 0 . First, fix such an x 1 ∈ Ω and define which is a C 1 function satisfying DF (x) = 0 for anyx (by (Twist)). In particular, F cannot attain its maximum over the compact set exp c x0 (∂u (x 0 )) except at the boundary, say atx 0 ∈ exp c x0 (∂u (x 0 ) ∂ ) ⊂ ∂ c u(x 0 ). Thus if there exists which is a violation of c-monotonicity of the c-subdifferential of u (see Remark 2.5).
As a result there cannot be such anx 1 , and we obtain (8). Since ν = T # µ, we must then have exp c x0 (∂u (x 0 ) int ) ∩ spt ν = ∅. However, when combined with (7) this exactly implies that exp c x0 (∂u (x 0 ) int ) is a hole in spt ν which contradicts our initial assumption, therefore it must be that affdim ∂u (x 0 ) < n.
If c also satisfies (A0), (Nondeg), and (MTW), by Corollary 2.8 we have that exp c x0 (∂u (x 0 ) int ) = ∂ c u (x 0 ) int and is c-convex with respect to x 0 ; the conclusion thus follows from the same proof as above.
In the next lemma, we extend to c-convex functions the following easy result about convex functions: if a convex function u makes contact with an affine function along a line segment containing a point x 0 , then either u is singular along the line segment or the gradient of the affine function is an exposed point of the convex set ∂u (x 0 ). Our extension is, in particular, to cost functions such that Loeper's maximum principle, Theorem 2.7 holds. Proof. Fix ap 0 that is not an extremal point of [∂ c u(x 0 )] x0 . There existp ± =p 0 such thatp ± ∈ [∂ c u(x 0 )] x0 andp 0 = 1 2 (p + +p − ); let us writex ± := exp c x0 (p ± ). Now, suppose by contradiction that there exists some x 1 ∈ S 0 with x 1 = x 0 . Consider the c-segment x(λ) := exp c x0 ((1 − λ)p 0 + λp 1 ), for λ ∈ [0, 1] from x 0 to x 1 ; observe from Corollary 2.9 that x(λ) ∈ S 0 for all λ ∈ [0, 1]. Also using that for all x ∈ Ω. In particular, for all λ ∈ [0, 1]. At the same time by using Theorem 2.7 (3), thus by combining these we must have the equality for all λ ∈ [0, 1]. Together with (9), this implies that for each λ ∈ [0, 1], either x + ∈ ∂ c u(x(λ)) orx − ∈ ∂ c u(x(λ)). Sincex + ,x − =x 0 by construction, and clearlȳ x 0 ∈ ∂ c u(x(λ)) for all λ ∈ [0, 1] this implies all points x(λ) in the c-segment must be singular points, contradicting that x 0 is an isolated singular point. This proves In order to prove the main theorem, we require a modified version of the estimate [12, Lemma 6.10] (this is proven in the same vein as [10, Proposition 1] for the Euclidean case of c(x,x) = − x,x ). By the notation |·| L , we denote the volume of a set in M ,M or an associated cotangent space, induced by the Riemannian metric on either M orM (which will be clear from context). x 0 from opposite sides, and let ℓ Π ± be the length of the longest line segment orthogonal to Π ± that is contained in ≥ δ. Then (writing p 0 := −Dc(x 0 ,x 0 )) there exists a constant C > 0 depending only on δ, n, Λ, diam (spt ν), and c such that Proof. First, one can use (2) and follow a proof analogous to [ where C > 0 depends on the cost function c. Now let K c x0,S0 (·) be the c-cone over the section S 0 with vertex x 0 . Then, by using [10, Lemma 3.1], we calculate where the final constant C depends on c, Λ, δ, and diam (spt ν). Combining this with the original proof of [12, Lemma 6.10], we immediately obtain the claim.
With all of the preceeding ingredients in hand, we are ready to prove the main theorem.
Proof of Theorem 1.1. Suppose by contradiction that u has an isolated singular point x 0 ∈ (spt µ) int . We begin by a localization of u around x 0 . [∂ c u(x 0 )] x0 is convex by Corollary 2.8 (4) and contains more than one point since u is singular at x 0 ; thus there must exist at least one non-extremal pointp 0 of [∂ c u(x 0 )] x0 . Let us define a family of sections around x 0 using c-affine functions with focusx 0 := exp c x (p 0 ), for h > 0 let S h := {x ∈ Ω | u(x) ≤ −c(x,x 0 ) + c(x 0 ,x 0 ) + u(x 0 ) + h} .
Notice that by Lemma 3.3, it holds the section is a singleton when h = 0, i.e. S 0 = {x 0 }. As a result S h can be made sufficiently small around x 0 for small enough h > 0. Thus by the assumption that x 0 is an isolated singularity, we may assume h > 0 to be small enough that S h ⊂ (spt µ) int and u is differentiable on On the other hand, by Proposition 3.2 we see that the affine dimension of ∂u (x 0 ) is strictly less than n, and in particular ∂u (x 0 ) = ∂u (x 0 ) ∂ . Hence by Proposition 3.1, the definition of Brenier solution, and closedness of spt ν, we see that In particular,x 0 ∈ spt ν. Since u is differentiable on S h \ {x 0 }, (10) and the definition of Brenier solution imply ∂ c u(S h ) ⊂ spt ν.
Now consider the c-cone K c x0,S h (x) over S h with vertex x 0 as in Lemma 2.10. From the condition spt ν ∩Ω ∂ = ∅, it holdsx 0 ∈Ω int , therefore we can apply Lemma 2.10 (5) and (6) to see > 0) in place of [12,Theorem 6.11], to obtain that u is differentiable at x 0 ; this contradicts that x 0 is a singular point, completing the proof.