Weak formulation of the MTW condition and convexity properties of potentials
aa r X i v : . [ m a t h . A P ] S e p WEAK FORMULATION OF THE MTW CONDITIONAND CONVEXITY PROPERTIES OF POTENTIALS
GR ´EGOIRE LOEPER , NEIL S TRUDINGER Monash University, School of Mathematical Sciences The Australian National University, Mathematical Sciences Institute
Abstract.
We simplify the geometric interpretation of the weakMa-Trudinger-Wang condition for regularity in optimal transporta-tion and provide a largely geometric proof of the global c-convexityof locally c -convex potentials when the cost function c is only as-sumed twice differentiable. Introduction
We consider a cost function c defined on the product Ω × Ω ∗ of twodomains Ω , Ω ∗ in Euclidean space R n . For a mapping φ : Ω → R wedefine its c-transform φ c : Ω ∗ → R by ∀ y ∈ Ω ∗ , φ c ( y ) = sup x ∈ Ω {− φ ( x ) − c ( x, y ) } . Conversely we define the c ∗ - transform of ψ : Ω ∗ → R . A c-convexpotential has at every point x ∈ Ω a c-support, i.e., there exists y ∈ Ω ∗ , ψ = ψ ( y ) ∈ R such that ∀ x ′ ∈ Ω , φ ( x ′ ) ≥ − ψ ( y ) − c ( x ′ , y ) , with equality at x ′ = x . It follows from this definition that φ ( x ) = sup y ∈ Ω ∗ {− ψ ( y ) − c ( x, y ) } and that φ can be obtained as the c ∗ transform of ψ : Ω ∗ → R . Itthen turns out that ψ = φ c . For φ a c-convex potential, and φ c itsc-transform, we define as in [2] the contact set as a set valued map G φ given by G φ ( y ) = { x : φ ( x ) + φ c ( y ) = − c ( x, y ) } . for y ∈ Ω ∗ . We will also use the notions of c-segment, c-convexity ofdomains. Whenever needed, we will refer to the conditions A1 , A2 , A3 , A3w that have been introduced in [5, 8]. One of the main featuresof this paper is that we will assume throughout that the cost function
E-mail address : [email protected], [email protected] . Date : September 9, 2020. is globally C (Ω × Ω ∗ ), without any further explicit smoothness hy-potheses. As usual we will use subscripts to denote partial derivativesof c with respect to variables x ∈ Ω and subscripts preceded by acomma to denote partial derivatives with respect to y ∈ Ω ∗ , so that inparticular c x , c i , c ,y , c ,j , c i,j denote the partial derivatives of c with re-spect to x, x i , y, y j , x i y j . We also use c x,y = [ c i,j ] to denote the inverseof the matrix c x,y = [ c i,j ]. We further assume throughout the paperthat c satisfies the assumptions A1 , A2 of [5], that is for all x ∈ Ω themapping y → − c x ( x, y ) is injective, that the dual counterpart holdsand the matrix c x,y is not singular. We also introduce what will be aweak form of assumption A3w : Definition 1.1.
The cost function satisfies
A3v if: for all x, x ∈ Ω and y , y ∈ Ω ∗ , for all θ ∈ (0 , , with c x ( x, y θ ) = θc x ( x, y ) + (1 − θ ) c x ( x, y ) , there holds max {− c ( x, y ) + c ( x , y ) , − c ( x, y ) + c ( x , y ) }≥ − c ( x, y θ ) + c ( x , y θ ) + o ( | x − x | ) , where the term o ( | x − x | ) may depend on θ . From [2] it is known that when the cost function is C , A3v isequivalent to
A3w .Our main result is the following:
Theorem 1.2.
Let c : Ω × Ω ∗ → R be a C cost-function satisfying A1 , A2 with Ω , Ω ∗ c-convex with respect to each other. Assume that (i) c satisfies A3v .Then (ii) for all y , y ∈ Ω ∗ , σ ∈ R , the set U = { x ∈ Ω : c ( x, y ) − c ( x, y ) ≤ σ } is c-convex with respect to y , (iii) for all φ c-convex, x ∈ Ω , y ∈ Ω ∗ , the contact set G φ ( y ) and itsdual G φ c ( x ) are connected, (iv) any locally c-convex function in Ω is globally c-convex. Remark.
The novelty of the result lies in the way it is obtained;at no point do we have to differentiate the cost function c . Hence thecomputations from previous proofs [1, 4, 7], in the case when c ∈ C ,do not have to be reproduced. The proof will be based on a purelygeometric interpretation of condition A3v .2.
Proof of Theorem 1.2
In what follows we will use the term c − exponential (c-exp), as in [2],to denote the mapping in condition A1 , that is y = c-exp x ( p ) ⇔ − c x ( x, y ) = p. e recall also that D p (c-exp x ) = − c x,y . The core of the proof lies in the following two lemmas,
Lemma 2.1 (c-hyperplane lemma) . Let x ∈ Ω , y , y ∈ Ω ∗ and let y θ = c-exp x p θ where p θ = (1 − θ ) c x ( x , y ) + θc x ( x , y ) , ≤ θ ≤ , denote a point on the c-segment from y to y , with respect to x .Consider, for θ > , the section, S θ = S ( x , y , y θ ) := { x ∈ Ω : c ( x, y ) − c ( x , y ) ≤ c ( x, y θ ) − c ( x , y θ ) } . Then as θ approaches 0, ∂S θ ∩ Ω converges to H , the c ∗ -hyperplanewith respect to y , passing through x , with c-normal vector p − p ,given by H = H ( x , y , y ) = { x ∈ Ω : − c x,y ( x , y )( p − p ) · [ c ,y ( x, y ) − c ,y ( x , y )] = 0 } Proof.
Locally around θ = 0, the equation of ∂S θ reads[ c ,y ( x, y ) − c ,y ( x , y )] · ( y θ − y ) = o ( θ ) . Passing to the limit as θ goes to 0, we obtain[ c ,y ( x, y ) − c ,y ( x , y )] · ∂ θ y θ = 0 , which gives the desired result, since ∂ θ y θ = − c x,y ( x , y )( p − p ) . (cid:3) Remark.
We call H a c-hyperplane with respect to y because ifwe express x as c ∗ -exp y ( q ) then H = c ∗ -exp y ( ˜ H ) , or equivalently ˜ H = − c ,y ( · , y )( H ) , where˜ H = { q ∈ c ,y ( · , y )(Ω) : c x,y ( x , y )( p − p ) . ( q − q ) = 0 } , q = − c ,y ( x , y )Therefore, H is the image by c ∗ -exp y of a hyperplane. Remark.
We will define in the same way, (replacing 0 by θ and θ by θ ′ ), the section S θ,θ ′ , for θ ′ ∈ ( θ, c ∗ -hyperplane, H θ =lim θ ′ → θ S θ,θ ′ .The following lemma is then the second main ingredient of the proof:it says that the c-convexity of S θ is non-decreasing with respect to θ ; (note that the previous lemma asserts that the c-convexity of S θ vanishes at θ = 0). emma 2.2. Assume that c satisfies A3v .Then the second fundamen-tal form of ∂S θ at x is non-decreasing with respect to θ , for θ in (0 , . Proof.
Consider h θ = c ( x, y ) − c ( x, y θ ) − c ( x , y ) + c ( x , y θ ) . Note that h θ is a defining function for S θ in the sense that S θ = { x ∈ Ω : h θ ≤ } .Note also that at x = x we have h θ ( x ) = 0 for all θ and the set { ∂ x h θ | x = x , θ ∈ [0 , } is a line. Therefore all the sets ∂S θ contain x and have the same unitnormal at x .Then we note that property A3v is equivalent to the following: lo-cally around x we have h θ ≤ max { h , } + o ( | x − x | ) . (1)(To see this, we just subtract c ( x , y ) − c ( x, y ) from both sides of theinequality A3v ).Then (1) implies that the second fundamental form of ∂S θ cannotstrictly dominate the second fundamental form of ∂S in any tangentialdirection at x . By changing y into y θ ′ for θ ′ ≥ θ , this implies that thesecond fundamental form of ∂S θ is non-decreasing with respect to θ . (cid:3) Remark.
We remark that analytically the conclusion of Lemma2.2 can be expressed as a co-dimension one convexity of the matrix A ( x, p ) = − c xx ( x, c-exp x ( p )) with respect to p , in the sense that thequadratic form Aξ.ξ is convex on line segments in p orthogonal to ξ ormore explicitly: h A ij ( x, p θ ) − (1 − θ ) A ij ( x, p ) − θA ij ( x, p ) i ξ i ξ j ≤ , (2)for all ξ ∈ R n such that ξ · ( p − p ) = 0, which, for arbitrary y , y ∈ Ω ∗ ,is clearly equivalent to A3w when c ∈ C .We now deduce assertion ( ii ) in Theorem 1.2 from A3v ; this will bedone in several steps.Step 1. Uniform boundedness of the section’s curvature (including c-hyperplanes)From the previous corollary, it follows that θ → c xx ( x , y θ ) ξ i ξ j is convexand therefore Lipschitz, and for a.e. θ ∈ [0 , A = ∂ θ c x i x j ( x , c-exp x ( p θ )) ξ i ξ j xists and is equal to lim θ ′ → θ B ( θ, θ ′ ) where B ( θ, θ ′ ) = (cid:0) c x i x j ( x , c-exp x ( p θ ′ ) − c x i x j ( x , c-exp x ( p θ ) (cid:1) ξ i ξ j θ ′ − θ . The first term A would be the curvature of H θ if it exists. The secondterm B in the limit is the curvature of S θ,θ ′ . We can deduce right awaythat the curvature of S θ,θ ′ remains uniformly bounded at x thanksto (2). Now this reasoning can be extended to any point x ∈ ∂S θ,θ ′ ,although the c-segment between y θ and y ′ θ will be with respect to x ,but the conclusion that the curvature of S θ,θ ′ at x is uniformly boundedremains. Therefore the curvature of all sections is uniformly boundedso as the uniform limit of ∂S θ,θ ′ , H θ is a C , hypersurface, and thereforehas a curvature a.e. given by A .Step 2. Local convexity Wherever A is well defined, the curvature of H θ is equal to A . Moreover, for θ ′ > θ , the second fundamental formof ∂S θ,θ ′ dominates a.e. the one of H θ .Let us define the hypersurfaces P m = { x ∈ Ω , c ( x, y ) − c ( x, y ) = m } , m ∈ R . By standard measure theoretical arguments, the previous result impliesthe following:
Lemma 2.3.
For a.e. y , y , m there holds at H n − every point x on P m ( y , y ) , that - the second fundamental form (SFF) of H ( x , y , y ) at x iswell defined, let us call it A , equivalently H ( x , y , y ) is twicedifferentiable (as a hypersurface) - A is dominated by the SFF of ∂S ( x , y )- going back to the tangent space (i.e. composing with c ,y ( · , y ) ),the second fundamental form of c ,y ( · , y )( ∂S ( x , y )) dominatesthe null form. We now conclude the local convexity. Starting from a point x where H and S are tangent. Both are defined by x , y , y . Representing ∂S and H as graphs over R n − , and we denote by s and s thecorresponding functions. We assume x = 0, and that both graphshave a flat gradient at 0. For x ′ ∈ R n − we have s i ( x ′ ) = | x ′ | Z ∂ νν s i ( θx ′ )(1 − θ ) dθ, i = 0 , , where ν is the appropriate unit vector. By the definition of H , at agiven point x ∗ = ( x ′ , h ( x ′ )), H is tangent to S ∗ = S ( x ∗ , y , y ∗ ) , y ∗ = c-exp[ x ∗ , − c x ( x ∗ , y )+ c x,y ( x ∗ , y ) c x,y ( x , y )( p − p )] . or almost every choice of x there will hold for a.e. x ′ that ∂ νν s ( x ′ ) ≤ ∂ νν s ∗ ( x ′ ) ≤ ∂ νν s ( x ′ ) + ε ( x ′ − ε = 0, depending on the continuity of c xx , c x,y . Therefore s ( x ′ ) ≤ | x ′ | ( Z ∂ νν s ( θx ′ )(1 − θ ) dθ + ε ( x ′ )) ≤ s ( x ′ ) + ε ( x ′ − x ) | x ′ | . Going now in the tangent space, for q ′ in a well chosen n − π the projection on { x n = 0 } , we call x ( q ′ ) = π (c ∗ -exp( y , q ′ )) andwe have s ( x ( q ′ )) ≤ s ( x ( q ′ )) + ε ( x ( q ′ ) − x ) | x ( q ′ ) | ,s ( x ( q ′ )) is an affine function, s ( x ( q ′ )) defines the image of ∂S by c ,y and ε ( x ( q ′ )) | x ( q ′ ) | ≤ ˜ ε ( q ′ )) | q ′ | for some ε ′ . For a.e. choice of x , thisholds for a.e q ′ . More importantly the ε ′ is (locally) uniform. Thisimplies the convexity through the following lemma Lemma 2.4.
Let s be C . Assume that for some continous ε ( · ) with ε (0) = 0 , there holds for almost every x , xs ( x ) ≥ l x ( x ) − ε ( x − x ) | x − x | l x being the tangent function at x , then s is convex.Proof. Elementary, both sides of the inequality are continuous in x, x , so this holds in fact everywhere. (cid:3) Remark.
For a proof of local convexity without using Lemma 2.3the reader is referred to [3].Global convexity To complete the proof of assertion (ii), we need toshow that the set ˜ S is connected. The proof goes as follows, and it isvery close to the argument of [8], Section 2.5. Let σ be a constant, andassuming that the set { c ( x, y ) − c ( x, y ) ≤ σ } has two disjoint components, we let σ increase until the two componentstouch in a C c -convex subdomain Ω ′ ⊂⊂ Ω. From the local convexityproperty this can only happen on the boundary of Ω ′ . At this point,say x there holds locally that c ( x, y ) − c ( x, y ) ≤ σ on ∂ Ω ′ and for x ε = x − εν , ν the outer unit normal to Ω ′ , c ( x ε , y ) − c ( x ε , y ) > h. his implies that c ( x, y ) − c ( x, y ) ≥ h is locally c-convex around x , a contradiction, and from this we deducethat S can have at most one component. Since a connected locallyconvex set in Euclidean space must be globally convex, we thus deducethat S is globally c-convex. (cid:3) An analytical proof for a smooth cost function.
If a C domain Ω is defined locally by ϕ >
0, its local c-convexity with respectto y , for c ∈ C , is expressed by h ϕ ij + c ij,k c k,l ( · , y ) ∂ l ϕ i τ i τ j ≥ , or equivalently h ϕ ij + ∂ p A ij .∂ϕ i τ i τ j ≥ τ ∈ ∂ Ω [7]. Plugging ϕ ( x ) = c ( x, y ) − c ( x, y ) − h into thisinequality, we obtain immediately from (2) that S is locally c -convexwith respect to y .More generally this argument proves Theorem 1.2when we assume additionally that the form Aξ.ξ is differentiable withrespect to p in directions orthogonal to ξ .2.2. Connectedness of the contact set.
This new characterizationimplies right away the c -convexity of the global c -sub-differential, ( c -normal mapping). We prove now that ( i ) implies ( iii ).For φ c-convex, we have φ ( x ) = sup y {− φ c ( y ) − c ( x, y ) } , (3) φ c ( y ) = sup x {− φ ( x ) − c ( x, y ) } . (4)Then { φ ( x ) ≤ − c ( x, y ) + h } = ∩ y { x : − φ c ( y ) − c ( x, y ) ≤ − c ( x, y ) + h } = ∩ y { x : c ( x, y ) ≤ c ( x, y ) − h + φ c ( y ) } . Therefore { φ ( x ) ≤ − c ( x, y ) + h } is an intersection of c-convex sets andhence also c-convex. We then have G φ ( y ) = { x, φ ( x ) = − c ( x, y ) − φ c ( y ) } = { x, φ ( x ) ≤ − c ( x, y ) − φ c ( y ) } , and hence G φ ( y ) is a c-convex set. To show the dual conclusion, we mayrewrite assertion (ii) as: for all y, y ∈ Ω ∗ , x , x ∈ Ω ∗ and θ ∈ (0 , c y ( x θ , y ) = θc y ( x , y ) + (1 − θ ) c y ( x , y ) , here holds max {− c ( x , y ) + c ( x , y ) , − c ( x , y ) + c ( x , y ) }≥ − c ( x θ , y ) + c ( x θ , y ) . Since this shows in particular that
A3v is invariant under duality wecomplete the proof of assertion (iii). Moreover as a byproduct of thisargument we also see that the sets S θ are non-increasing with respectto θ and that A3v holds without the term o ( | x − x | ). (cid:3) Local implies global.
We prove that ( ii ) implies ( iv ). We con-sider φ a locally c-convex function, i.e, φ has at every point a localc-support. Locally, φ can be expressed as φ ( x ) = sup y ∈ ω {− ψ ( y ) − c ( x, y ) } , for some ω ( x ) ⊂ Ω ∗ (if φ was globally c-convex there would hold that ω ≡ Ω ∗ and ψ would be equal to φ c ). It follows that the level sets S m,y = { x : φ ( x ) + c ( x, y ) ≤ m } are locally c-convex with respect to y for any y . We obtain that − ∂ y c ( S m,y , y ) is locally convex. Reasoning again as in the proof of theglobal convexity in point ( ii ) (i.e. increasing m until two componentstouch), we obtain that, for φ locally c-convex, − ∂ y c ( S m,y , y ) is globallyconvex for all y . This implies in turn the global c-convexity of φ ,following Proposition 2.12 of [2]. As already mentioned, this part isvery similar to the argument of [7], section 2.5.Finally we remark that the arguments in this paper extend to gen-erating functions as introduced in [6] and also provide as a byproductan alternative geometric proof of the invariance of condition A3w un-der duality to the more complicated calculation in [6]. The resultantconvexity theory is presented in [3]. (cid:3)
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