Optimal transport maps on Alexandrov spaces revisited

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Optimal transport maps on Alexandrov spaces revisited © 2021 the Authors Published version Rajala, Tapio; Schultz, Timo Rajala, T., & Schultz, T. (2022). Optimal transport maps on Alexandrov spaces revisited. Manuscripta Mathematica, 169(1-2), 1-18. https://doi.org/10.1007/s00229-021-01333-3 2022 manuscripta math. © The Author(s) 2021 Tapio Rajala ·Timo Schultz Optimal transport maps on Alexandrov spaces revisited Received: 31 March 2020 / Accepted: 4 August 2021 Abstract. We give an alternative proof for the fact that in n-dimensional Alexandrov spaces with curvature bounded below there exists a unique optimal transport plan from any purely (n−1)-unrectifiable starting measure, and that this plan is induced by an optimal map. Our proof does not rely on the full optimality of a given plan but rather on the c-monotonicity, thus we obtain the existence of transport maps for wider class of (possibly non-optimal) transport plans. 1. Introduction The problem of optimal mass transportation has a long history, starting from the work of Monge [34] in the late 18th century. In the original formulation of the problem, nowadays called the Monge-formulation, the problem is to find the transport map Tminimizing the transportation cost Rn c(x,T(x)) dμ0(x), (1.1) among all Borel maps T:Rn→Rntransporting a given probability measure μ0to another given probability measure μ1, that is, Tμ0=μ1. In the original problem of Monge, the cost function c(x,y)was the Euclidean distance. Later, other cost functions have been considered, in particular much of the study has involved the distance squared cost, c(x,y)=|x−y|2, which is the cost studied also in this paper. In the Monge-formulation (1.1) of the optimal mass transportation problem the class of admissible maps Tthat send μ0to μ1is in most cases not closed in any suitable topology. To overcome this problem, Kantorovich [26,27] considered a larger class of optimal transports, namely, measures πon Rn×Rnsuch that the first marginal of πis μ0and the second is μ1. Such measures πare called transport Both authors partially supported by the Academy of Finland, Grants no. 274372, 312488, and 314789 Tapio Rajala (B)·Timo Schultz Department of Mathematics and Statistics, University of Jyvaskyla, P.O. Box 35 (MaD), 40014 Jyvaskyla, Finland. e-mail: [email protected]; [email protected] Mathematics Subject Classification: Primary 53C23 ·Secondary 49K30 https://doi.org/10.1007/s00229-021-01333-3 T. Rajala, T. Schultz plans. Kantorovich’s relaxation leads to the so-called Kantorovich-formulation of the problem, inf πRn×Rn c(x,y)dπ(x,y). (1.2) Due to the closedness of the admissible transport plans and the lower semicontinuity of the cost, minimizers exist in the Kantorovich-formulation under very mild assumptions on the underlying space and the cost c. For the quadratic cost in the Euclidean space, it was shown independently by Brenier [9] and Smith and Knott [42] that having μ0absolutely continuous with respect to the Lebesgue measure guarantees that the optimal transport plans (minimizer of (1.2)) are unique and given by a transport map. Moreover, the optimal transport map is given by a gradient of a convex function. The results of Brenier and of Smith and Knott have been generalized in many ways. The most important directions of generalization have been: going from the underlying space Rnto other metric spaces, considering other cost functions, and relaxing the assumption of the starting measure being absolutely continuous with respect to the reference measure (here the Lebesgue measure). In this paper, we study the direction of relaxing the absolute continuity in a more general metric space setting, the Alexandrov spaces. We note that one should be able to generalize our proof for more general costs, such as the distance to a power p∈(1,∞).In order to keep the presentation simpler, we concentrate here on the distance squared cost. The existence of optimal transportation maps in Alexandrov spaces with curvature bounded below for starting measures that are absolutely continuous with respect to the reference Hausdorff measure was proven by Bertrand [6]. Later Bertrand improved this result [7] by relaxing the assumption on the starting measure to give zero measure to c−c-hypersurfaces. Here we provide an alternative proof for the result of Bertrand under the slightly stronger assumption on the starting measure of pure (n−1)-unrectifiability (see Definition 2.1 for the definition of pure (n−1)-unrectifiability). Theorem 1.1. Let (X,d)be an n-dimensional Alexandrov space with curvature bounded below. Then for any pair of measures μ0,μ 1∈P2(X)such that μ0is purely (n−1)-unrectifiable, every c-monotone plan πfrom μ0to μ1is induced by a map. In particular, there exists a unique optimal transport plan from μ0to μ1and this transport plan is induced by a map. Remark 1.2. (1) The uniqueness of the optimal transport plan follows from the fact that Opt(μ0,μ 1)is a convex subset of the set of c-monotone plans. (2) While the motivation for the above formulation of Theorem 1.1 arises from the optimal mass transportation theory, it could be restated in the spirit of regularity of monotone operators, cf. [46,48]. The contribution of this paper is to provide a different approach to showing the existence and uniqueness of optimal transport maps than what was used by Optimal transport maps on Alexandrov spaces revisited Bertrand in [6,7]. In [6], Bertrand used the local (1+ε)-biLipschitz maps to Rn on the regular set of X, and the general existence of Kantorovich potentials and their Lipschitzness. Since the singular set of Xis at most (n−1)-dimensional, and the Rademacher’s theorem on Rncan be restated in Xvia the biLipschitz maps, Bertrand concluded that the optimal transport is concentrated on a graph that is given by applying the exponential map to the gradient of the Kantorovich potential. In [7], Bertrand considered the problem in boundaryless Alexandrov spaces. He used Perelman’s DC calculus to translate the problem to differentiability of convex functions on Euclidean spaces. Then the result follows from the characterization of nondifferentiability points of convex functions due to Zajíˇcek [47]. In this paper, we translate a contradiction argument (Lemma 2.11)fromthe Euclidean space (which uses just monotonicity in certain geometric configurations) to the space Xvia the (1+ε)-biLipschitz charts. In order to use the contradiction argument, we need to get all the used distances to be comparable. For this we use the fact that the directions of geodesics are well-defined in the biLipschitz charts (Theorem 2.7) and thus we can contract along the geodesics without changing the geometric configuration too much. Finally, the geometric configurations that result in the contradiction via cyclical monotonicity are given by the pure (n−1)- unrectifiability (Lemma 2.2). Let us briefly describe the contradiction argument in the Euclidean case X=R2under the assumption that the starting measure μ0is absolutely continuous with respect to the Lebesgue measure. Suppose towards a contradiction that we have an optimal transport πtransporting μ0to μ1so that πis not induced by a map. Then, after some discretizations, we find a positive measure set Aof points where πtransports measure to two different directions that are roughly some directions v1and v2. Then, since Ahas positive μ0-measure and μ0 is absolutely continuous, in a Lebesgue point Xof Athere is another point ynearby roughly in the direction v1−v2from x. But now, the lines from xto the direction v1and from yto the direction v2cross. Such crossing violates the optimality of πbecause by interchanging the endpoints of the transports corresponding to the intersecting lines, we would decrease the cost of π. To the best of our knowledge, in the context of optimal transportation this contradiction argument was first used by Champion, De Pascale and Juutinen [16] to prove the existence of optimal maps for the ∞-transportation distance. Similar idea was also used by Champion and De Pascale [14] to solve the Monge problem in Rd. The limits of the contradiction argument were later pushed further by Champion and De Pascale [15] and by Jylhä [25]. Let us comment also on the history of the sufficient assumptions on μ0.The assumption of pure (n−1)-unrectifiability was shown by McCann [33]tobesufficient for the existence of optimal maps in the case of Riemannian manifolds. A sharper condition based on the characterization by Zajíˇcek [47] of the set of nondifferentiability points of convex functions was first used in the Euclidean context by Gangbo and McCann [20] when they showed that having an initial measure that gives zero mass to c−c-hypersurfaces is sufficient to give the existence of optimal maps. It was then shown by Gigli [21] that even in the Riemannian manifold context the sharp requirement for the starting measure to have optimal maps for any target measure is indeed that it gives zero measure to c−c-hypersurfaces. T. Rajala, T. Schultz It still remains open whether zero measure of c−c-hypersurfaces also gives a full characterization in the case of Alexandrov spaces. One of the directions, the sufficiency, was obtained by Bertrand [7]. The existence of optimal maps has been studied in wider classes of metric measure spaces that satisfy some form of Ricci curvature lower bounds or weak versions of measure contraction property. These classes include CD(K,N)-spaces that were introduced by Lott and Villani [32], and by Sturm [43,44], MCP(K,N)- spaces (see Ohta [35]), and RCD(K,N)spaces that were first introduced in the case N=∞by Ambrosio, Gigli and Savaré [3] and then for general Nby Gigli [23] (see also the improvements and later work by Ambrosio, Gigli, Mondino and Rajala [1], Erbar, Kuwada and Sturm [18] and Ambrosio, Mondino and Savaré [4]). All of these classes contain Alexandrov spaces with curvature lower bounds, see Petrunin [37]. It was first shown by Gigli [22], that in nonbranching CD(K,N)-spaces you do have the existence of optimal maps provided that the starting measure is absolutely continuous with respect to the reference measure. In all the subsequent work, the assumption has been the same for the starting measure, and it would be interesting to see if it can be relaxed also in the more general context of metric measure spaces with Ricci curvature lower bounds. Also a metric version of Brenier’s theorem was studied by Ambrosio, Gigli and Savaré [2]. They did not obtain the existence of optimal maps, but showed that at least the transportation distance is given by the Kantorovich potential. Later, Ambrosio and Rajala [5] showed that under sufficiently strong nonbranching assumptions one can conclude the existence of optimal maps. Rajala and Sturm [39] noticed that strong CD(K,∞)spaces, and hence RCD(K,∞)spaces are at least essentially nonbranching, and that this weaker form of nonbranching is sufficient for carrying out Gigli’s proof. This result was later improved by Gigli, Rajala and Sturm [24]. Essential nonbranching was then studied together with the measure contraction property MCP(K,N)by Cavalletti and Mondino [13] (see also Cavalletti and Huesmann [12] where the case of nonbranching and a weaker version of MCP(K,N)was considered), and finally it was shown by Kell [29] that under a weak type measure contraction property, the essential nonbranching characterizes the uniqueness of optimal transports and that the unique optimal transport is given by a map for absolutely continuous starting measures. The role of nonbranching and measure contraction type properties was also studied by De Pascale and Rigot [17] in connection with their sollution of the Monge problem in the Heisenberg group. See also the work of Bianchini and Cavalletti [8] on the Monge problem in nonbranching geodesic spaces. The existence of optimal transport maps in CD(K,N)spaces without any extra assumption on nonbranching is still an open problem. An intermediate definition between CD(K,N)and essentially nonbranching CD(K,N), called very strict CD(K,N), was studied by Schultz [40]. He showed that in these spaces one still has optimal transport maps even if the space could be highly branching and the optimal plans non-unique. It is also worth noting that if one drops the assumption of essential nonbranching for MCP(K,N), then optimal transport maps need not exist. This is seen from the examples by Ketterer and Rajala [30]. Optimal transport maps on Alexandrov spaces revisited The paper is organized as follows. In Sect. 2we recall basic things about rectifiability, Alexandrov spaces and optimal mass transportation. While doing this, we also present a few facts that easily follow from well-known results: purely n−1-unrectifiable measures have mass in all directions (Lemma 2.2), the singular set in an Alexandrov space is (n−1)-rectifiable (Theorem 2.5), gradients of geodesics exist in charts in Alexandrov spaces (Theorem 2.7) and the failure of cyclical monotonicity persists after small perturbations (Lemma 2.11). In Sect. 3 we then put these things together and prove Theorem 1.1. 2. Preliminaries In this paper (X,d)always refers to a complete and locally compact length space. By a length space we mean a metric space where the distance between any two points xand yis equal to the infimum of lengths of curves connecting xand y. By the Hopf-Rinow-Cohn-Vossen Theorem, our spaces (X,d)are then geodesic, proper and, in particular, separable. A space is called geodesic, if any two points in the space can be connected by a geodesic. By a geodesic we mean a constant speed length minimizing curve γ:[0,1]→X. Notice that we parametrize all the geodesics by the unit interval. We denote the space of geodesics of Xby Geo(X)and equip it with the supremum-distance. By a (geodesic) triangle (x,y,z)we mean points x,y,z∈Xand any choice of geodesics [x,y],[y,z]and [x,z]pairwise connecting them. 2.1. Rectifiability For our Theorem 1.1 the starting measure μ0is diffused enough if it is purely n−1-unrectifiable. Let us recall this notion. Definition 2.1. AsetA⊂Xis called (countably) k-rectifiable if there exist Lipschitz maps fi:Ei→Xfrom Borel sets Ei⊂Rkfor i∈N, such that A⊂i∈Nfi(Ei). A measure μis called purely k-unrectifiable,ifμ(A)=0 for every k-rectifiable set A. The property of purely unrectifiable measures that we use is that they have mass in all directions. This is made precise using (one-sided) cones that are defined as follows. Given x∈Rn,θ∈n−1,α>0 and r>0, we denote the open cone at xin direction θwith opening angle α,by C(x,θ,α):= y∈Rn:y−x,θ>cos(α)|y−x|. Lemma 2.2. Let μbe a purely (n−1)-unrectifiable measure on Rnand let E ⊂Rn with μ(E)>0. Then at μ-almost every x ∈E we have C(x,θ,α)∩B(x,r)∩E=∅ for all θ∈n−1,α>0and r >0. T. Rajala, T. Schultz Proof. Suppose that there is a subset E0⊂Ewith μ(E0)>0 such that the conclusion fails, i.e. for every x∈E0there exist θx∈n−1,αx>0 and rx>0 such that C(x,θ x,α x)∩B(x,rx)∩E=∅. Since C(x,θ,α)∩B(x,r)⊂C(x,θ,α)∩B(x,r) if α≥αand r≥r, there exist r>0 and α>0 such that the subset {x∈E0:C(x,θ x,α)∩B(x,r)∩E=∅} has positive μ-measure. By considering a countable dense set of directions {θi}i∈N, we have that there exists one fixed direction θisuch that the set E1:= {x∈E0:C(x,θ i,α/2)∩B(x,r)∩E=∅} has positive μ-measure. But now, for evey x∈Rn,thesetE1∩B(x,r/2)is contained in a Lipschitz graph and hence E1is an (n−1)-rectifiable set, giving a contradiction with the pure (n−1)-unrectifiability of μ. 2.2. Alexandrov spaces Let us recall some basics about Alexandrov spaces. Unless we provide another source, all the following definitions and results can be found in [10]. Alexandrov spaces generalize sectional curvature bounds by means of comparison to constant curvature model spaces. Alexandrov spaces can be defined for instance by comparing geodesic triangles of a metric space to the corresponding ones in a model space. Let us next give precise definitions. For each k∈R,letMkbe a simply connected surface with constant sectional curvature equal to k, that is, for negative k,Mkis a scaled hyperbolic plane, for k=0, Mkis the Euclidean plane, and for positive k,Mkis a (round) sphere. Let us denote the distance between two points x,y∈Mkby |x−y|. Let k∈R. For a triplet x,y,z∈X,let ˜x,˜y,˜z∈Mkbe points so that the triangles (x,y,z)and ( ˜x,˜y,˜z)have the same side lengths, that is, d(x,y)= |˜x−˜y|,d(y,z)=|˜y−˜z|,d(x,z)=|˜x−˜z|. We call the triangle ( ˜x,˜y,˜z)a comparison triangle for (x,y,z). For a triangle (x,y,z)in Xwe denote by ˜ k(y,x,z)the comparison angle at ˜xin the comparison triangle ( ˜x,˜y,˜z)in Mk. Definition 2.3. (Alexandrov space) We say that (X,d)is an Alexandrov space (with curvature bounded below by k) if there exists k∈Rso that for each point p∈X there exists a neighbourhood Uof pfor which the following holds. If (x,y,z)⊂ U,( ˜x,˜y,˜z)its comparison triangle in Mk, and w∈[x,y],˜w∈[˜x,˜y]with d(x,w)=|˜x−˜w|, then d(w, z)≥|˜w−˜z|. An Alexandrov space might have infinite (Hausdorff) dimension. In this paper we study only finite dimensional Alexandrov spaces. Recall that in an Alexandrov space every open nonempty set has the same dimension, so the dimension of an Alexandrov space is always well defined. Moreover, the dimension is either an Optimal transport maps on Alexandrov spaces revisited integer or infinity. From now on, the space (X,d)isassumedtobeann-dimensional Alexandrov space with curvature bounded below by k∈Rwith n∈N. We will use the fact that our purely (n−1)-unrectifiable starting measures μ0 live on the regular set of the space, that has nice charts. Let us recall the notion of regular and singular points. Definition 2.4. A point p∈Xis called regular, if the space of directions pat p is isometric to the standard sphere n−1, or equivalently, if the Gromov-Hausdorff tangent at pis the Euclidean Rn. A point p∈Xthat is not regular is called singular. The set of regular points of Xis denoted by Reg(X)and the set of singular points by Sing(X). The following result is from [36](seealso[11]). It implies that our starting measures μ0give zero measure to the singular set. Theorem 2.5. The set Sing(X)is (n−1)-rectifiable. Proof. Notice that [36, Theorem A] states that Sing(X)has Hausdorff dimension at most n−1. However, the proof easily gives the stronger conclusion of (n−1)- rectifiability. Namely, observe that in the proof of [36, Theorem A] Otsu and Shioya show that Sing(X)is contained in Lipschitz images from subsets of the spaces of directions pfor countably many points p∈X. Since the points pare only needed to locally form a maximal ε-discrete net in X, they can be chosen to be regular points of X. Thus, Sing(X)is contained in countably many Lipschitz images from subsets of n−1and is therefore (n−1)-rectifiable.  Let us then recall a well-known consequence of the nonbranching property of Alexandrov spaces. For its proof, we need the notion of an angle. Let α, β :[0,1]→Xbe two constant speed geodesics emanating from the same point p=α(0)=β(0). Let us denote by θk(t,s):= ˜ k(α(t), p,β(s)) the angle at ˜pof the comparison triangle ( ˜p,˜α(t), ˜ β(s)) in Mkof (p,α(t), β(s)). In Alexandrov spaces the angle (α, β) := lim t,s0θk(t,s) is well-defined for every pair of geodesics α, β emanating from the same point. Moreover, by Alexandrov convexity (see for instance [41, Sect. 2.2]) the quantity θk(t,s)is monotone non-increasing in both variables tand s. Lemma 2.6. Let γ1,γ 2:[0,1]→X be be two constant speed geodesics with γ1(0)=γ2(0)and γ1(1)= γ2(1). Then lim t0 d(γ1(t), γ2(t)) t>0. Proof. We may assume (γ1)≥(γ2).If(γ1) > (γ2), then by triangle inequality d(γ1(t), γ2(t)) ≥t((γ1)−(γ2)), giving the claim. If (γ1)=(γ2), then θk(1,1)=˜ k(γ1(1), x,γ 2(1)) > 0. Then by Alexandrov convexity, (γ1,γ 2)≥ θk(1,1)>0, and thus by the cosine law d(γ1(t), γ2(t)) t→(γ1)2−2 cos((γ1,γ 2)) > 0, T. Rajala, T. Schultz as t→0.  Our aim is to arrive at a contradiction with cyclical monotonicity at a small scale near a regular point. We will transfer the Euclidean argument to the Alexandrov space Xusing the following standard charts ϕ. Since we need the existence of directions of geodesics in these charts, we write the existence down explicitly inside the following theorem. Theorem 2.7. For every p ∈Reg(X)and every ε>0there exist a neighborhood U of p and a (1+ε)-biLipschitz map ϕ:U→Rnwith ϕ(U)open so that for every constant speed geodesic γ:[0,1]→U the limit lim t0 ϕ(γ(t)) −ϕ(γ(0)) d(γ (t), γ (0)) exists. Proof. We recall (see [36]or[10, Theorem 10.8.4]) that the local (1+ε)-biLipschitz chart ϕ:U→Rncan be obtained as ϕ(x)=(d(a1,x), d(a2,x),...,d(an,x)), where (ai,bi)n i=1is a δ-strainer for p,forsomeδ>0. Now, the first variation formula (see [36, Theorem 3.5] or [10, Theorem 4.5.6, Corollary 4.5.7]) implies that lim t0 d(ai,γ(t)) −d(ai,γ(0)) d(γ (t), γ (0)) =−cos(α), where α=(γ, β), with βa geodesic from γ(0)to ai. Thus, the required limit exists for each i. 2.3. Optimal mass transportation In this section we recall a few basic things in optimal mass transportation. The Monge–Kantorovich formulation of optimal mass transportation problem (with quadratic cost) is to investigate for two Borel probability measures μ0and μ1the following infimum inf X×X d2(x,y)dπ(x,y), where the infimum is taken over all Borel probability measures π∈P(X×X) which has μ0and μ1as a marginals, that is, π(A×X)=μ0(A)and π(X×A)= μ1(A)for all Borel sets A∈B(X). In order to guarantee that the above infimum is finite, it is standard to assume the measures μ0and μ1to have finite second moments. The set of all Borel probability measures in Xwith finite second moments is denoted by P2(X). An admissible measure that minimizes the above infimum is called an optimal (transport) plan, and the set of optimal plans between μ0and μ1is denoted by Optimal transport maps on Alexandrov spaces revisited Using the definition of the set B, and (3.4), (3.3) and (3.6), we have 1 (1+ε)2d2(x2,γ2 x1(t)) ≤|ϕ(γ2 x1(t)) −ϕ(x2)|2 =|1 2t(y1+y2)+(ϕ(γ 2 x1(t)) −ϕ(x1)−ty2)−(ϕ(x2)−ϕ(x1)−1 2t(y2−y1))|2 ≤|1 2t(y1+y2)|+|ϕ(γ2 x1(t)) −ϕ(x1)−ty2|+|ϕ(x2)−ϕ(x1)−1 2t(y2−y1)|2 ≤|1 2t(y1+y2)|+tˆε+1 2|t(y2−y1)|ˆε2 ≤|1 2t(y1+y2)|+(3k+1)tˆε2 ≤|1 2t(y1+y2)|2+6tk(3k+1)tˆε+((3k+1)tˆε)2≤|1 2t(y1+y2)|2+40t2k2ˆε and similarly 1 (1+ε)2d2(x1,γ2 x2(t)) ≤| 1 2t(y1+y2)|2+40t2k2ˆε. Thus, by summing the two terms, using (3.2) and the fact that x1∈Ak, 1 (1+ε)2[d2(x2,γ2 x1(t)) +d2(x1,γ1 x2(t))] ≤2|1 2t(y1+y2)|2+80t2k2ˆε ≤1 2|t(y1+y2)|2+t2 k2ε≤1 2|t(y1+y2)|2+εd2(γ 2 x1(t), x1). (3.7) Again, by the definition of the set Band the choice of ˆε |ty1|2≤(1+ε)d(γ 1 x2(t), x2)+tˆε2≤(1+ε)d(γ 1 x2(t), x2)+εd(γ 2 x1(t), x1)2 ≤((1+ε)2+2(1+ε)ε)d2(γ 1 x2(t), x2)+(ε2+2(1+ε)ε)d2(γ 2 x1(t), x1) ≤(1+7ε)d2(γ 1 x2(t), x2)+5εd2(γ 2 x1(t), x1)(3.8) and |ty2|2≤(1+ε)d(γ 2 x1(t), x1)+tˆε2 ≤(1+2ε)2d2(γ 2 x1(t), x1)≤(1+8ε)d2(γ 2 x1(t), x1). (3.9) Using the inequalities (3.5), (3.8) and (3.9), we get that 1 2|t(y1+y2)|2<(1−δ)(|ty2|2+|ty1|2) ≤(1−δ)(1+13ε)(d2(γ 2 x1(t),x1)+d2(γ 1 x2(t),x2)). (3.10) T. Rajala, T. Schultz Hence, by (3.7), (3.10), the fact that δ≤1 2and the choice of ε, we have that d2(x2,γ2 x1(t)) +d2(x1,γ1 x2(t)) ≤(1+ε)21 2|t(y1+y2)|2+εd2(γ 2 x1(t), x1) ≤(1+ε)2(1−δ)(1+15ε)(d2(γ 2 x1(t), x1)+d2(γ 1 x2(t), x2)) ≤(1−δ)(1+100ε)(d2(γ 2 x1(t), x1)+d2(γ 1 x2(t), x2)) <d2(x2,γ1 x2(t)) +d2(x1,γ2 x1(t)). However, since (x2,γ1 x2(1)), (x1,γ2 x1(1)) ∈spt(π) we have by Lemma 2.10 that d2(x2,γ1 x2(t)) +d2(x1,γ2 x1(t)) ≤d2(x2,γ2 x1(t)) +d2(x1,γ1 x2(t)). which is a contradiction. Therefore, the plan πis induced by a map. Acknowledgements The authors thank Jérôme Bertrand, Luigi De Pascale, Nicola Gigli, and the anonymous referee for their comments on the earlier versions of this paper. Funding Open access funding provided by University of Jyväskylä (JYU). Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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