CONTINUITY AND INJECTIVITY OF OPTIMAL MAPS

CONTINUITY AN INJECTIVITY OF OPTIMA MAPS JÉRÔME VÉTOIS Abstract. Figalli Kim McCann proved in [14] the continuity and injectivity of optimal maps under the assumption B3 of nonnegative cross-curvature. In the recent [15, 16], they extend their results to the assumption A3w of Trudinger-Wang [34], and they prove, moreover, the Hölder continuity of these maps. We give here an alternative and independent proof of the extension to A3w of the continuity and injectivity of optimal maps based on the sole arguments of [14] and on new Alexandrov-type estimates for lower bounds. 1. Introduction Given a cost function c : R n R n R, n, and two probability densities f and g in R n with respect to ebesgue s measure, Monge s problem of optimal transportation consists in finding a minimizer of the cost functional C T := c x, T x f x dx R n among all measurable maps T : R n R n pushing the measure with density f forward to the measure with density g i.e. g = f for all Borel subsets Γ of Γ T 1 Γ Rn. A solution of the optimal transportation problem is called an optimal map. Figalli Kim McCann proved in [14] a continuity and injectivity result for optimal maps under the assumption B3 of nonnegative cross-curvature of the cost function extending the works by oeper [9] and Trudinger Wang [35] where the result was proved under the stronger condition A3. In the recent [15, 16], Figalli Kim McCann extend their results to the assumption A3w of Trudinger Wang [34], and they prove, moreover, the Hölder continuity of optimal maps. In this paper, we give an alternative proof which was found independently of the extension to A3w of the continuity and injectivity of optimal maps based on the sole arguments of [14] and on new Alexandrov-type estimates for lower bounds see Theorem 3.3 and Corollary 3.5. These estimates are, on some aspects, more general than the estimates obtained recently by Figalli Kim McCann [15]. Our proof relies, in particular, on invariance properties of the cost function under affine renormalization, properties which are observed but not used in [15]. The derivation of new estimates for lower bounds turns out to be the main difficulty in the extension of the regularity of optimal maps to A3w. In the rest of the proof, we follow, and slightly adapt when necessary, the very nice ideas by Figalli Kim McCann [14]. The strategy of the proof had been developed originally by Caffarelli [4] for the Monge Ampère equation. Several tricky obstacles arise when applying this strategy to more general cost functions. These obstacles have been overcome by Figalli Kim McCann [14 16] and, alternatively, in this paper as regards the extension from B3 to A3w of the continuity and injectivity. ate: July 7, 11. Revised: November 13, 13. Published in Calculus of Variations and Partial ifferential Equations 5 15, no. 3, 587 67. Most of this work has been done during the visit of the author at the Australian National University. The author was supported by the Australian Research Council and by the CNRS-PICS grant Progrès en analyse géométrique et applications. 1

CONTINUITY AN INJECTIVITY OF OPTIMA MAPS In this paper, we assume A A3w below for a cost function c : U R in a bounded open subset U of R n R n. A c C 4 U. A1 For any x, y U and for any p, q x c x, U x y c U y, y, where U x := {y R n ; x, y U} and U y := {x R n ; x, y U}, there exist unique Y = Y x, p and X = X q, y such that x c x, Y = p and y c X, y = q. A For any x, y U, det xyc x, y. A3w For any x, y U and ξ, η R n R n such that ξ η, there holds ppa x, p. ξ, ξ, η, η, 1.1 where Y x, p = y, i.e. p = x c x, y, and A x, p := xxc x, Y x, p. The assumptions A A3w are the same assumptions under which Trudinger Wang [34] obtained a smoothness result for optimal maps when the densities are smooth. See also the former reference by Ma Trudinger Wang [3] where the result was proved under assumption A3 which consists in asking that the inequality in 1.1 is strict. In the same context of smooth densities, oeper [9] proved that A3w is necessary for the continuity of optimal maps. For more general measures, under the assumption A3, oeper [9] proved the Hölder continuity of optimal maps, and Trudinger Wang [35] gave a different proof for the continuity these results follow from more general regularity results for potential functions proved in [9] and [35]. iu [6] improved the result of oeper [9] by obtaining a sharp Hölder exponent. Figalli Kim McCann, in their first paper on the question [14], proved the continuity and injectivity of optimal maps under the following assumption B3 of nonnegative cross-curvature now called B4 in [15]. B3 For any curve t [ 1, 1] y c x t, y, x c x, y t which is an affinely parametrized line segment, there holds cross x,y [x, y ] := 4 s t c x s, y t. 1. s,t=, A3w is equivalent to asking that 1. holds provided s t s,t=, c x s, y t =. We refer to Kim McCann [4, 5] for a detailed discussion on the notion of cross-curvature. In addition to our assumptions on the cost function, we assume that the densities lie in two open subsets Ω + and Ω of R n which satisfy B below. B Ω + Ω U, Ω + and Ω are strongly c-convex with respect to each others. Ω + and Ω are said to be c-convex resp. strongly c-convex with respect to each others if x c x, Ω and y c Ω +, y are convex resp. strongly convex for all x, y Ω + Ω. A convex set K is said to be strongly convex if there exists R > such that for any x K, K B x Rν R for some outer unit normal vector ν to a supporting hyperplane of K at x, where B x Rν R is the ball of center x Rν and radius R. In case K is smooth, K is strongly convex if and only if all principal curvatures of its boundary are bounded below by 1/R. Strong convexity is also called uniform convexity in Trudinger Wang [34, 35]. Our main result states as follows. We refer to Section and to the references therein for discussions on Kantorovich duality. Theorem 1.1. Assume that the cost function satisfies A A3w. et Ω, Ω +, and Ω be three bounded open subsets of R n such that Ω Ω + and such that B holds. et f 1 Ω + and g 1 Ω be two probability densities such that f Ω +, 1/f Ω, g Ω, and 1/g Ω. et T : Ω + Ω be the optimal map determined by Kantorovich duality. Then T is continuous and one-to-one in Ω.

CONTINUITY AN INJECTIVITY OF OPTIMA MAPS 3 Regularity of optimal maps has been intensively studied. Historic references on this question for the cost c x, y = x y are by Brenier [3], Caffarelli [4 9], elanoë [1], and Urbas [36]. We also mention the related works by Wang [39, 4] on the reflector antenna design problem. Another early result of regularity by Gangbo McCann [] addressed the case of the squared cost when restricted to the product of the boundaries of two strongly convex sets. For more general cost functions, without pretension of exhaustivity, recent advances on the regularity of optimal maps are by elanoë Ge [11, 1], Figalli Kim McCann [13 16], Figalli oeper [17], Figalli Rifford [18], Figalli Rifford Villani [19, ], Kim McCann [4, 5], iu [6], iu Trudinger[7], iu Trudinger Wang [8], oeper [9, 3], oeper Villani [31], Ma Trudinger Wang [3], and Trudinger Wang [33 35]. More references can be found in Figalli Kim McCann [14, 15]. We refer to [14, 15] for an extensive discussion on the problem of regularity of optimal maps. We also refer to Gangbo McCann [1], Urbas [37], and to the book by Villani [38] where one can find more general material on optimal transportation. The paper is organized as follows. We begin with some preliminaries in Section. In particular, in Section, we state the regularity results for potential functions which imply Theorem 1.1. In Sections 3 and 4, we derive Alexandrov-type estimates for lower and upper bounds. We combine this material in Section 5 in order to prove the regularity of potential functions. As an appendix, we state some results on optimal transportation and convex sets. Acknowledgments: The author is very grateful to the Australian National University for its generous hospitality during the period when the majority of this work has been carried out. The author wishes to express his gratitude to Philippe elanoë, Neil Trudinger, and Xu-Jia Wang for supporting his visit to Canberra, and for several stimulating discussions on optimal transportation. The author is indebted to Neil Trudinger for having introduced him to the problem of regularity of optimal maps. The author wishes also to express his gratitude to Emmanuel Hebey for helpful support and advice during the redaction, and to Jiakun iu and, again, Neil Trudinger for their valuable comments and suggestions on the manuscript.. Preliminaries.1. Potential functions. We let U, Ω +, Ω, c satisfy A A3w, B, and we let Ω be an open subset of Ω +. By Kantorovich duality, it is known, see for instance Caffarelli [9], Gangbo McCann [1], and Villani [38], that the solution of the optimal transportation problem is almost everywhere uniquely determined by a map T : Ω + Ω satisfying the relation x c x, T x = u x a.e. in Ω +.1 for some c-convex function u in Ω + focussing in Ω, i.e. such that there exists v : Ω R satisfying v y = u c y and u x = v c x for all x, y Ω + Ω, where u c y := sup x Ω + c x, y u x and v c x := sup y Ω c x, y v y.. The functions u and v are said to be potential functions for the optimal transportation problem. We refer, for instance, to Kim McCann [4] and to the book by Villani [38] for more material on c-convex functions. In particular, c-convex functions are semiconvex, i.e. x u x + C x is locally convex in Ω + for some C >. We define the c-subdifferential of u at x Ω + by c u x := { y Ω ; u x + u c y + c x, y = },.3 where u c is as in.. The function u is differentiable at some point x Ω if and only if c u x is a singleton, and in this case, we get c u x = {T x}, where T x is as in.1. In particular, T is continuous and one-to-one in Ω if and only if u is continuously differentiable

CONTINUITY AN INJECTIVITY OF OPTIMA MAPS 4 and strictly c-convex in Ω, i.e. for any y c u Ω, there exists a unique x Ω such that y c u x. Moreover, see Figalli Kim McCann [14, emma 3.1] see also Ma Trudinger Wang [3], we get c u f/g 1 Ω Ω n in Ω and c u g/f Ω + Ω n in Ω +, in the sense of measures, where n is ebesgue s measure and c u Γ = n c u Γ for all Borel subset Γ of Ω +. In particular, in order to prove Theorem 1.1, we can assume that the potential function u satisfies C and below. C u is c-convex in Ω + focussing in Ω. There exist Λ 1, Λ > such that c u Λ 1 n in Ω and c u Λ n in Ω + in the sense of measures, where n is the ebesgue measure in R n. In Theorems.1 and. below, we state our regularity results for potential functions. The proofs of Theorems.1 and. are left to Section 5. Theorem.1. Assume that U, Ω, Ω +, Ω, c, u satisfy A A3w, B, C, and. Then u is strictly c-convex in Ω. Theorem.. Assume that U, Ω, Ω +, Ω, c, u satisfy A A3w, B, C, and. Then u is continuously differentiable in Ω. Proof of Theorem 1.1. As discussed above, Theorem 1.1 follows from Theorems.1 and. by Kantorovich duality... Convexity of sublevel sets. We let U, Ω +, Ω, c, u satisfy A A3w, B, C,, and we let Ω be an open subset of Ω +. We fix y Ω. We define U := { y c x, y, y ; x, y U and x, y U}, Ω := y c Ω, y, and Ω + := y c Ω +, y..4 Clearly, Ω is an open subset of Ω +. We define a new cost function c in U by c q, y := c X q, y, y c X q, y, y,.5 and we define a new function u in Ω + by u q := u X q, y + c X q, y, y,.6 where X q, y is as in A1. As is easily checked, after the above change of coordinates, U, Ω, Ω +, c, u satisfy the same conditions as U, Ω, Ω +, c, u, except A which is replaced by Aw below. In other words, U, Ω +, c, u satisfy Aw A3w, B, C, and. Aw c C 3 U and 4 qqyyc exists and is continuous in U. Moreover, c satisfies two additional conditions, namely that q c q, y = and y c q, y = q.7 for all q Ω +. In particular, it follows from B and.7 that Ω + is strongly convex. In emma.3 below, we state the property of convexity of sublevel sets. This property has been proved and used independently by Figalli Kim McCann [14] and iu [6]. emma.3 is stated under the assumption Bw below, slightly weaker than B. Bw Ω + Ω U, Ω + and Ω are c-convex with respect to each others. emma.3. et U, Ω +, Ω, c, u satisfy A A3w, Bw, C, and let y Ω. et U, Ω +, c, u be as in.4.6. Then for any q, q Ω +, there holds u [q, q ] max u q, u q. In particular, u has convex sublevel sets, i.e. for any R, the set is convex. Ω := { q Ω + ; u q }.8

CONTINUITY AN INJECTIVITY OF OPTIMA MAPS 5 3. Alexandrov-type estimates for lower bounds We let U, Ω +, Ω, c, u satisfy A A3w, Bw, and C. We fix y Ω, and we let U, Ω +, c, u be as in.4.6. Moreover, for any compact convex subset K of Ω +, we define { σ c, y, K, Ω pp A q, p. ξ, ξ, η, η := sup ; ξ, η q [q, q + ] K, p [, η], η q c q, Ω, and ξ = q + q }, 3.1 where A q, p := qqc q, Y q, p and q c q, Y q, p = p. We adopt the convention that σ c, y, K, Ω = in case diam K =. Preliminary to this section, we prove the following lemma. emma 3.1. et U, Ω +, Ω, c satisfy A A3w, Bw, and let y Ω. et U, Ω +, c be as in.4 and.5. Then for any compact convex subset K of Ω +, there holds σ c, y, K, Ω pp A qc diam K 3. where σ c, y, [q, q ], Ω and A are as in 3.1, diam K and diam Ω are the diameters of K and Ω. Proof. We get the first inequality in 3. by letting ξ in the definition of σ c, y, K, Ω. Now, we prove the second inequality. We let q, q + K, q [q, q + ], ξ = q + q, η q c q, Ω, and p [, η]. By A3w, we get ξ, η ppa q, p. ξ, ξ, η, η ppa q, p. η η, ξ, η, η + ppa q, p. ξ ξ, η ξ, η η, η η η, η, η pp A ξ, η ξ η. 3.3 Since K is convex and q, q + K, we get [q, q + ] K. Moreover, we get η q c q, Ω. It follows that η q c and ξ diam K. 3.4 The second inequality in 3. then follows from 3.3 and 3.4. Our next lemma states as follows. emma 3.. et U, Ω +, Ω, c satisfy A A3w, Bw, and let y Ω. et U, Ω +, c be as in.4 and.5. Then for any q, q Ω + and y Ω such that q c q, y. q q, there holds q c q, y. q q e σc,y,[q,q ],Ω / q c q, y. q q, 3.5 where σ c, y, [q, q ], Ω is as in 3.1. Proof. For any t [, 1], we define ϕ t := c q t, y, where q t := 1 t q +tq. By.7, we get qqc q t, y = and 3 qqyc q t, y =. By A A3w, Bw, and the mean value theorem, it follows that there exists s [, 1] such that ϕ t = qqc q t, y. q q, q q = 1 ppa q t, sp. q q, q q, p, p 1 σ c, y, [q, q ], Ω q q, p = 1 σ c, y, [q, q ], Ω ϕ t, 3.6

CONTINUITY AN INJECTIVITY OF OPTIMA MAPS 6 where Y q t, p = y, i.e. q c q t, y = p, and A q t, sp is as in 3.1. Integrating 3.6, we get that either ϕ t < for all t [, 1] or ϕ and ϕ t e σc,y,[q,q ],Ω t/ ϕ for all t [, 1]. In particular, in case t = 1 and ϕ 1, we get 3.5. In what follows, we often combine emma 3. with the mean value theorem. oing so, we get that for any q, q Ω + and y Ω, if q c q, y. q q, then q c q, y. q q e σc,y,[q,q ],Ω / c q, y c q, y. 3.7 Similarly, we get that for any q, q Ω + and y Ω, if c q, y c q, y, then c q, y c q, y e σc,y,[q,q ],Ω / q c q, y. q q. 3.8 We state our general Alexandrov-type estimates for lower bounds in Theorem 3.3 below. In case the cost function satisfies the assumption B3 of nonnegative cross-curvature, see Figalli Kim McCann [14], estimates for lower bounds can be deduced from the fact that in this case, the cost measure is dominated by the Monge Ampère measure. Our estimates state as follows. Theorem 3.3. et U, Ω +, Ω, c, u satisfy A A3w, Bw, C, and let y Ω. et U, Ω +, c, u be as in.4.6. et R be such that Ω and Ω Ω +, where Ω is as in.8. Then for any K Ω such that K, there holds inf u n C det 1 qy c 1 e nσc,y,ω,ω d K, Ω K d K, Ω n c u K 3.9 + diam K for some C = C n >, where σc, y, Ω, Ω is as in 3.1, diam K is the diameter of K, and d K, Ω is the distance between K and Ω. Proof. For any q K, we define E q := { y Ω ; c q, y c q, y u q q Ω }. Since u = on Ω, we have c u q E q. We claim that for any q, q K, and y E q, there holds q c q, y e σc,y,ω,ω u q d K, Ω + diam K d K, Ω. 3.1 We prove this claim. Since Ω is bounded and q K Ω, we get that there exists q Ω such that q c q, y. q q = q c q, y q q. By 3.7, it follows that q c q, y q q e σc,y,[q,q ],Ω / c q, y c q, y. 3.11 Since y E q and q Ω, we get c q, y c q, y c q, y c q, y + u q. 3.1 If q = q, then it follows directly from 3.11 and 3.1 that q c q, y eσc,y,[q,q ],Ω / q q n u q eσc,y,ω,ω / d u K, Ω q, 3.13

CONTINUITY AN INJECTIVITY OF OPTIMA MAPS 7 and thus we get 3.1. Therefore, we can assume that q q. Since Ω convex, we get that there exists a unique q Ω such that q q q q = q q q q. By 3.7 and 3.8, it follows that if c q, y c q, y, then is bounded and c q, y c q, y e σc,y,[ q,q ],Ω / q q q q c q, y c q, y. 3.14 Since y E q and q Ω, we get c q, y c q, y u q. 3.15 By 3.11 3.15, we get q c q, y q q e σc,y,ω,ω u q 1 + q q. 3.16 q q Our claim 3.1 follows from 3.16 in view of q q d K, Ω, q q d K, Ω and q q diam K. In particular, by 3.1, we get that q c q, E K B R = B 1 R n, 3.17 where R := e σc,y,ω,ω inf u d K, Ω + diam K K d K, Ω. Moreover, since c u q E q for all q K, we get c u K E K det qyc 1 q c q, E K. 3.18 Finally, 3.9 follows from 3.17 and 3.18. In what follows, we combine Theorem 3.3 with affine renormalizations. The invariance under affine renormalization of the optimal transportation problem with general costs have been observed but not used in Figalli Kim McCann [15]. We let : R n R n be an affine transformation such that det. We define := det /n t 1, where t 1 and det are, respectively, the transpose of the inverse and the determinant of the linear part of. In particular, we have det = det. For simplicity, we denote det := det. We define U := { x, 1 y ; x, y U }, Ω := 1 Ω, c x, y := det /n c x, y, and u x := det /n u x. 3.19 As is easily checked, U, Ω, c, and u satisfy A A3w, Bw, C. Moreover, given y Ω, we define y := 1 y, and similarly to.4.6, we define U := { y c x, y, y ; x, y U and x, y U }, Ω + c q, y := c X q, y, y c X q, y, y, := y c Ω +, y, u q := u X q, y + c X q, y, y, Ω := { q Ω + ; u q }, 3.

CONTINUITY AN INJECTIVITY OF OPTIMA MAPS 8 where := det /n and X q, y is such that y c X q, y, y = q. By direct calculations, we get X q, y = X q, y, and thus U = { 1 q, 1 y ; q, y U }, Ω + = 1 Ω +, c q, y = det /n c q, y, Now, we prove the following key property of σc, y, Ω, Ω. u q = det /n u q, and Ω = 1 Ω. 3.1 emma 3.4. et U, Ω +, Ω, c satisfy A A3w, Bw, and let y Ω. et U, Ω +, c be as in.4.5, and U,, Ω + c be as in 3. 3.1. For any affine transformation : R n R n and any compact convex subset K of Ω +, there holds where K := 1 K. Proof. We fix We define σ c, y, K, Ω = σ c, y, K, Ω, 3. q [q, q + ] K, ξ := q + q, η q c q, Ω, and p [, η]. 3.3 q := 1 q, q ± := 1 q ±, ξ := q + q, η := 1 η, and p := 1 p. 3.4 By 3.1, 3.3, and 3.4, we get q [ q, q +] K, ξ = q + q, η q c q, Ω, and p [, η ]. 3.5 Now, we claim that ppa q, p. ξ, ξ, η, η = ppa q, p. ξ, ξ, η, η, 3.6 ξ, η ξ, η where A q, p := qqc q, Y q, p and q c q, Y q, p = p. From 3.1, we derive Y q, p = 1 Y q, p, q, p. ξ, ξ = det /n A q, p. ξ, ξ, and thus ppa Moreover, we have A q, p. ξ, ξ, η, η = det /n ppa q, p. ξ, ξ, η, η. 3.7 ξ, η = det /n ξ, η. 3.8 Our claim 3.6 then follows from 3.7 and 3.8. Finally, we deduce 3. from 3.5 and 3.6. By combining Theorem 3.3 and emma 3.4, we get Corollary 3.5 below. We use this result in Section 5 in order to prove Theorems.1 and.. Corollary 3.5. et U, Ω +, Ω, c, u satisfy A A3w, Bw, C, and let y Ω. et U, Ω +, c, u be as in.4.6. et R be such that Ω Ω + and the interior of Ω is nonempty, where Ω is as in.8. et E be a n-dimensional ellipsoid such that E Ω ne, where ne is the dilation of E with respect to its center of mass the existence of such an ellipsoid is given by Theorem 6.5. Assume that there exist Λ 1 >, δ, 1, and

CONTINUITY AN INJECTIVITY OF OPTIMA MAPS 9 q Ω such that c u Λ 1 n in E δ := q c E + δe in the sense of measures, where n is the ebesgue measure in R n and c E is the center of E. Then there holds inf Ω u n CΛ1 det qy c 1 1 e nσc,y,ω,ω δ n Ω for some C = C n >, where σc, y, Ω, Ω is as in 3.1. 3.9 Corollary 3.5 can be compared with the estimates obtained recently by Figalli Kim McCann [15]. Under similar conditions as in Corollary 3.5, Figalli Kim McCann [15, Theorem 6.] prove that, provided that δ < ε c / 4diam E, where ε c > depends on the cost function, there holds inf Ω u n CΛ1 det qy c 1 1 δn Ω 3.3 for some C = C n >. Clearly, for δ small, the lower bound in 3.3 is better than ours in 3.9 because of the exponential term which depends on the cost function in 3.9. For δ fixed and diam E large, the lower bound in 3.9 is better since 3.3 is established provided that δ < ε c / 4diam E. In general, the question whether our estimate or Figalli Kim McCann s one is better depends on δ, c, and diam E. Proof. By translation, we can assume that c E =. We let : R n R n be a linear transformation satisfying E = B 1, where B 1 is the ball of center and radius 1. We define y := 1 y, and we let U,, Ω + c, u,, and Ω be as in 3. 3.1. Since E Ω ne, we get B 1 Ω B n. We define K δ := { q Ω E δ ; d 1 q, Ω } δ/ n, Kδ := 1 K δ = { q Ω Bq δ ; d q, Ω } δ/ n, where q := 1 q. By Theorem 3.3, we get inf u n Kδ C det qy c 1 1 e nσc,y,ω,ω d Kδ, Ω n d K δ, Ω c + diam K n u K δ δ 3.31 for some C = C n >. By definition of Kδ, we get d Kδ, Ω δ/ n and diam Kδ δ, and thus d Kδ, Ω d Kδ, δ Ω + diam K n 1 + 4n. 3.3 δ By direct calculations, we get inf Kδ u n inf Ω u n = det inf Ω u n, 3.33 det = Ω Ω Ω B n, 3.34 det qy c 1 = det 1 qy c, 3.35 c u K δ = 1 c u Kδ = det 1 c u Kδ. 3.36 Since K δ E δ, by assumption, we get c u Kδ Λ1 Kδ = Λ1 det K δ. 3.37 We claim that K Cδ n δ 3.38

CONTINUITY AN INJECTIVITY OF OPTIMA MAPS 1 for some C = C n >. We prove this claim. By convexity of Ω and since B 1 Ω and q, Ω we have Hull B 1 { q } Ω, where Hull E is the convex hull of a set E. By definition of Kδ, it follows that B q δ Hull B 1 δ { q δ } n n q K q δ. 3.39 Using that q n, we then deduce 3.38 from 3.39 by direct computations. Finally, 3.9 follows from 3.31 3.38 and emma 3.4. 4. Alexandrov-type estimates for upper bounds We let U, Ω +, Ω, c, u satisfy A A3w, Bw, and C. We fix y Ω, and we let U, Ω +, c, u be as in.4.6. Contrary to the estimates for lower bounds in the previous section, the extension to A3w of the Alexandrov-type estimates for upper bounds relies on a simple adaptation of the arguments by Figalli Kim McCann [14]. Except for this adaptation located in 4.7 4.8 in the proof of Theorem 4., and though we write the proofs in a slightly more direct way, we follow in this section the very nice ideas by Figalli Kim McCann [14]. We rewrite their arguments for the sake of completeness in our paper. Preliminary to the estimates for upper bounds, Figalli Kim McCann [14] establish the following ipschitz estimate. emma 4.1. et U, Ω +, Ω, c satisfy A A, Bw, and let y Ω. et U, Ω +, c be as in.4 and.5. Then for any q, q Ω + and y Ω, there holds q c q, y q c q, y β c 1 q q q c q, y, 4.1 where β c 1 = 3 qqy c 1 qyc. Proof. By the mean value theorem, we get that there exists t [, 1] such that q c q, y q c q, y = qqc q t, y. q c q, y q c q, y, q q, where q t = 1 t q + tq. It follows from.7 that q c q, y =, qqc q t, y =, and 3 qqyc q t, y =. By Bw and the mean value theorem, we then get that there exists s [, 1] such that q c q, y q c q, y = 3 qqyc q t, Y q, sp. q c q, y q c q, y, q q, p Y q, sp.p 3 qqy c py q c q, y q c q, y q q p, 4. where Y q, p = y, i.e. q c q, y = p. In particular, we get p Y = qyc 1. 4.1 then follows from 4.. Given a family of mutually orthogonal hyperplanes Π 1,..., Π n supporting a compact convex subset K of R n, we get the existence of a unique dual family of mutually orthogonal hyperplanes Π 1,..., Π n supporting K such that Π i//π i and Π i Π i for all i = 1,..., n. We let Box Π1,...,Π n K be the compact subset of R n delimited by the hyperplanes Π 1,..., Π n and Π 1,..., Π n. We state a general result on Alexandrov-type estimates for upper bounds in Theorem 4. below. In particular, Theorem 4. extends the original work by Alexandrov [1] which was concerned with the special case of the cost function c x, y = x y. Here, the nonlinearity of the modified cost makes the general situation much trickier than in [1]. In

CONTINUITY AN INJECTIVITY OF OPTIMA MAPS 11 the proof of Theorem 4., we mostly follow Figalli Kim McCann [14], except for the slight adaptation in 4.7 4.8 which extends the result to A3w. Theorem 4.. et U, Ω +, Ω, c, u satisfy A A3w, Bw, C, and let y Ω. et U, Ω +, c, u be as in.4.6. et R, q Ω, and Π 1,..., Π n be a family of mutually orthogonal hyperplanes supporting Ω. Assume that Box Π1,...,Π n Ω Ω +. There exists δ n depending only on n such that if diam Ω δn β c, where β c is as in emma 4.1 and diam Ω is the diameter of Ω, then u q n C n det cu qy c d q, Π i Ω 4.3 for some C = C n >, where d q, Π i is the distance between q and Π i. Proof. First, we claim that E := { y Ω ; c q, y c q, y u q q Ω i=1 } c u Ω in the notations of Figalli Kim McCann [14], E is the c -subdifferential at q of the c -cone generated by q and Ω with height u q. Indeed, since u = on Ω, we get that if y belongs to E, then the function v,y : q u q + c q, y c q, y u q is nonnegative on Ω. Since v,y q =, q Ω, and Ω is bounded, it follows that v,y admits a local minimum at some point q,y in Ω. By Corollary 6.3 in appendix, we then get that q,y is a global minimum point of v,y in Ω +, i.e. y c u q,y c u Ω. This ends the proof of 4.4. Now, for any i = 1,..., n, we choose q i Π i Ω and we claim that there exists y i E such that q c q i, y i is normal to Π i and 4.4 c q, y i c q i, y i = u q. 4.5 Indeed, by Corollary 6., we get u q i = q c q i, c u q i. In particular, by semiconvexity of u, since Ω is a level set of u and Π i is a supporting hyperplane of Ω, we get that there exists y i c u q i such that q c q i, y i is normal to Π i. Since y i c u q i, q i Ω, and u = on Ω, we get c q, y i c q i, y i u q 4.6 for all q Ω +. By Bw, by continuity of the function y c q, y c q i, y on the c - segment connecting y and y i with respect to q i, and since, by.7, c q, y c q i, y =, we get that there exists y i Ω on this c -segment such that 4.5 holds. By 4.6, since u = on Ω, and since, by.7, c q i, y c q, y = for all q Ω, by Theorem 6.1 in appendix, we get c q i, y i c q, y i for all q Ω. By 4.5, we then get y i E. Moreover, since q c q i, y i is normal to Π i and y i belongs to the c -segment connecting y and y i with respect to q i, we get that q c q i, y i is normal to Π i. By Theorem 6.1, we get that the function q c q, y i has convex sublevel sets. By 4.5 and since y i E, q i Π i, and q c q i, y i is normal to Π i, it follows that c q, y i c q, y i u q 4.7 for all q Π i. We let q i Π i be such that q q i = d q, Π i. Since Box Π1,...,Π n Ω Ω +, we get q i Ω +. By 4.7 and the mean value theorem, we get that there exists t [, 1] such that q c q t, y i. q q i u q, where q t = 1 t q i + tq. Since q q i = d q, Π i, it follows that q c q t, y i u q d q, Π i. 4.8

Moreover, we get CONTINUITY AN INJECTIVITY OF OPTIMA MAPS 1 q t q = 1 t q q i = 1 t d q, Π i d q, q i diam Ω. 4.9 It follows from 4.9 and emma 4.1 that q c q t, y i 1 + β c 1 diam Ω q c q, y i. 4.1 By emma 4.1, we also get q c q i, y i q c q i, y i qc q, y i q c q, y i qc q i, y i q c q, y i β c 1 q i q q c q, y i β c 1 diam Ω. 4.11 By continuity of the determinant function, we get that there exists a constant δ n depending only on n such that for any families e = e 1,..., e n and f = f 1,..., f n of vectors in R n, if e is orthogonal and e i f i e i f i δn for all i = 1,..., n, then det f 1 f 1 f f n. By 4.11, it follows that if diam Ω δn β c, then n q c q, y i det q c q, y i i=1,...,n = n! Hull q c q, y i. i=1,...,n 4.1 i=1 By Theorem 6.1 in appendix, we get that E is c -convex with respect to q, and thus q c q, E Hull q c q, y i i=1,...,n. 4.13 Moreover, by 4.4, we get c u Ω E det qyc 1 q c q, E. 4.14 Finally, 4.3 follows from 4.8 4.14. Now, still following Figalli Kim McCann [14], we fix a supporting hyperplane Π of Ω, and we claim that we can choose a family of mutually orthogonal hyperplanes Π 1,..., Π n supporting Ω such that Π 1 = Π and such that for any q Ω, there holds n d q, Π i CH n 1 π 1 Ω 4.15 i= for some C = C n >, where π 1 Ω is the projection of Ω onto Π. Indeed, in order to get 4.15, we can choose, for instance, the hyperplanes Π,..., Π n to be orthogonal to the axes of a n 1-dimensional ellipsoid E satisfying E π 1 Ω n 1 E which existence is given by Theorem 6.5. We then get the following corollary of Theorem 4. by applying Theorem 6.6 as in Figalli Kim McCann [14]. Corollary 4.3. et U, Ω +, Ω, c, u satisfy A A3w, Bw, C, and let y Ω. et U, Ω +, c, u be as in.4.6. et R, q Ω, and Π 1,..., Π n, Π 1 = Π, be a family of mutually orthogonal hyperplanes supporting Ω and such that 4.15 holds. Assume that Box Π1,...,Π n Ω Ω +. If diam Ω δn β c, where δ n and β c are as in emma 4.1 and Theorem 4. and diam Ω is the diameter of Ω, then u q n C det qy c d q, Π l Π Ω c u Ω 4.16 for some C = C n >, where d q, Π is the distance between q and Π, and l Π is the maximal length among all segments obtained by intersecting Ω with an orthogonal line to Π.

Proof. By 4.15 and Theorem 6.6, we get CONTINUITY AN INJECTIVITY OF OPTIMA MAPS 13 l Π n i= d q, Π i C Ω 4.17 for some C = C n >. 4.16 then follows from 4.17 and Theorem 4.. 5. Proofs of Theorems.1 and. We let U, Ω +, Ω, c, u satisfy A A3w, B, C, and we let Ω be an open subset of Ω +. We prove Theorems.1 and. by combining the Alexandrov-type estimates for lower and upper bounds in Corollaries 3.5 and 4.3. The proofs of Theorems.1 and. follow a strategy developed by Caffarelli [4] for the Monge Ampère equation and extended by Figalli Kim McCann [14] to the more general and more difficult setting of nonlinear cost functions. Proof of Theorem.1. We assume by contradiction that there exists y c u Ω such that c u 1 y Ω is not a singleton. By Theorem 6.4 in appendix and since c u 1 y is closed and Ω + is open and bounded, we get y Ω and c u 1 y Ω +. We let U, Ω, Ω +, c, u be as in.4.6. By emma.3, we get that c u 1 y is convex. Translating, if necessary, the set c u 1 y, we may assume that is an exposed point of c u 1 y. Since c u 1 y Ω is not a singleton, it follows that there exists q c u 1 y Ω \ {} and v c u 1 y \ {} such that v is normal to a supporting hyperplane of c u 1 y at. We define K := { q c u 1 y ; q, v q, v }, K := { q c u 1 y ; q, v γ v }, for some γ, 1 to be chosen small. In particular, we get K K. Moreover, since v c u 1 y \ {} is normal to a supporting hyperplane of c u 1 y at, we get q, v for all q K. For ε > small, letting y ε = y εv, we define U ε := { y c q, y ε, y ; q, y U and q, y ε U}, Ω ε := y c Ω, y ε, Ω + ε := y c Ω +, y ε, qε := y c q, y ε, q ε := y c γ v, y ε, and q ε := y c, y ε. By.7 and since c C 1 U and y ε y, we get Ω ε Ω, Ω + ε Ω +, q ε q, q ε γ v, and q ε. We define a new cost function c ε in U ε by and we define a new function u ε in Ω + ε c ε q, y = c X q, y ε, y c X q, y ε, y ε, by u ε q = u X q, y ε + c X q, y ε, y ε, where X q, y ε is as in A1. Moreover, we define K ε := { q Ω + ε ; u ε q u ε q ε }, K ε := { q Ω + ε ; u ε q u ε q ε }. By emma.3, we get that K ε and K ε are convex. We claim that K ε K and K ε K as ε. Indeed, we get { } y c ε K ε, y = q Ω + ; u q u q + c q, y ε c q, y ε. 5.1

For all q Ω +, by.7, we find CONTINUITY AN INJECTIVITY OF OPTIMA MAPS 14 c q, y ε c q, y ε 3 = y c q, y y c q, y. y ε y + O qyyc q q y ε y 3 = ε q q, v + O qyyc diam Ω + ε v. 5. It follows from 5.1 and 5. that y c ε K ε, y K as ε. In particular, since K c u 1 y Ω +, we get y c ε K ε, y Ω + for ε small. Since c C 1 U, y ε y, we get y c ε, y 1 = y c, y ε y c, y = id Ω + uniformly in Ω + as ε. It follows that K ε K as ε. In particular, since K c u 1 y Ω + and Ω ε + Ω +, we get K ε Ω ε + for ε small. Similarly, we get K ε K as ε. By definition of K ε and K ε, we get either K ε K ε or K ε K ε. Since their limits satisfy K K, we get K ε K ε for ε small. By Theorem 6.5, we get that there exists a n-dimensional ellipsoid E ε such that E ε K ε ne ε, where ne ε is the dilation of E ε with respect to its center of mass. Since K ε K as ε and K is bounded, we get that the diameter ot E ε is uniformly bounded for ε small. Since q ε q and Ω ε Ω as ε, and since q Ω and Ω is open, it follows that there exists δ, 1 such that E ε,δ := q ε c Eε + δe ε Ω ε for ε small, where c Eε is the center of E ε. In particular, by, we get that for any Borel subset Γ of E ε,δ, there holds cε u ε Γ = c u X Γ, y ε Λ 1 X Γ, y ε Λ 1 det q X 1 1 Γ = Λ 1 det xy c 1 Γ. By Corollary 3.5, it follows that n uε q ε inf u ε CΛ1 det xy c 1 K ε det 1 qy c ε 1 e nσcε,yε,kε,ω δ n K ε 5.3 for some C = C n >, where σc ε, y ε, K ε, Ω is as in 3.1. By 3.7, we get σ c ε, y ε, K ε, Ω pp A ε qc ε diam K ε, 5.4 where A ε q, p := qqc ε q, Y ε q, p and q c ε q, Y ε q, p = p. Moreover, we get det qy c ε 1 det xy c 1 det q X 1 = det xy c 1 det xy c, 5.5 q c ε x c q X = x c xy c 1, 5.6 pp A ε pp A q X q X 1 = pp A xy c 1 xy c. 5.7 By 5.3 5.7 and since diam K ε diam K < as ε, we get n uε q ε inf u ε C Kε. 5.8 K ε for some C > independent of ε. Since, γ v, q c u 1 y, by 5., and since q, v for all q K, we get u ε q ε u ε q ε u ε q ε inf Kε u ε c γ v, y ε c, y ε sup q yc εk ε,y c q, y ε c q, y ε γ v q, v + O 3 qyy c diam Ω + ε v. 5.9

CONTINUITY AN INJECTIVITY OF OPTIMA MAPS 15 Now, we end the proof by applying Theorem 4. and showing a contradiction with 5.8 and 5.9. For ε small, we define Π ε := { q R n ; q, v = inf q K ε q, v }. We let Π 1,ε,..., Π n,ε be a family of mutually orthogonal hyperplanes supporting K ε, such that Π 1,ε = Π ε, and such that 4.15 holds with Ω := K ε. Since Ω + is an exposed point of c u 1 y, since v c u 1 y \ {} is normal to a supporting hyperplane of c u 1 y at, and since K ε K, Ω ε + Ω +, and β c ε β c as ε, we can choose γ small enough so that Box Π1,ε,...,Π n,ε K ε Ω ε + and diam K ε < δ n β c ε, where β c ε and δ n are as in emma 4.1 and Theorem 4.. By Corollary 4.3, we then get uε q ε u ε q ε n C det qy c ε d q ε, Π ε l Πε K ε cε u ε K ε 5.1 for some C = C n >, where d q ε, Π ε is the distance between q ε and Π ε, and l Πε is the maximal length among all segments obtained by intersecting K ε with a line spanned by the vector v. By, we get cε u ε K ε = c u X K ε, y ε Λ X K ε, y ε Moreover, we get Λ det q X K ε Λ det xy c 1 K ε. 5.11 det qy c ε det xy c det q X det xy c det xy c 1. 5.1 It follows from 5.1 5.1 that uε q ε u ε q ε n d q C ε, Π ε K l ε 5.13 Πε for some C > independent of ε. By 5.8, 5.9, 5.13, and since K ε K ε, we get d q ε, Π ε Cl Πε 5.14 for some C > independent of ε. Since K ε K as ε, we get Π ε Π and l Πε l Π. Since q ε Π and l Π γ v, we then get a contradiction by passing to the limit as ε into 5.14. This ends the proof of Theorem.1. Now, using the strict c-convexity of u, we prove the continuous differentiability of u as follows. Proof of Theorem.. By Corollary 6. in appendix, we get that u x = x c x, c u x for all x Ω. In particular, by semiconvexity of u, in order to prove that u is continuously differentiable in Ω, it suffices to prove that for any x Ω, c u x is a singleton. By contradiction, we assume that there exists x Ω such that c u x contains at least two distinct points. We let y c u x be such that x c x, y is an exposed point of u x. We let U, Ω, Ω +, c, u be as in.4.6. We get u q = u x + x c x, y. q X q, y, 5.15 where X q, y = x. In particular, we get that is an exposed point of u q. Since c u x contains at least two distinct points, it follows that there exists v u q \ {} such that v is normal to a supporting hyperplane of u q at. For ε > small, we define { } K ε := q Ω + ; u q u q + ε.

CONTINUITY AN INJECTIVITY OF OPTIMA MAPS 16 By continuity of u and since, by Theorem.1, u is strictly c-convex, we get K ε {q } as ε. For ε small, we define Π ε := { q R n ; q, v = sup q, v q K ε }. Applying Corollaries 3.5 and 4.3 in the same way as in the proof of Theorem.1, we then prove that d q, Π ε Cl Πε 5.16 for some C > independent of ε, where l Πε is the maximal length among all segments obtained by intersecting K ε with a line spanned by the vector v. Now, we end the proof by estimating d q, Π ε and l Πε as ε and showing a contradiction with 5.16. On the one hand, by 3.7, letting y c u q be such that q c q, y = v, we get that for any q Ω +, if q, v q, v, then u q u q c q, y c q, y e σc,y,ω +,Ω / q c q, y. q q It follows that for ε small, there holds { } K ε q R n ; q q, v < εe σc,y,ω +,Ω /. In particular, we get = e σc,y,ω +,Ω / q q, v. d q, Π ε εe σc,y,ω +,Ω / / v. 5.17 On the other hand, since v u q \ {} is normal to a supporting hyperplane of u q at and since u is semiconvex, by the max formula, see Borwein ewis [, Corollary 6.1.], we get u q γv u q γ max p, v + o γ = o γ p u q as γ. It follows that for ε small, there exists γ ε > such that q γ ε v K ε and γ ε /ε. In particular, we get l Πε /ε, which contradicts 5.16 and 5.17. This ends the proof of Theorem.. 6. Appendix In this appendix, we gather some results on optimal transportation and convex sets that we use in this paper. 6.1. The maximum principle for cost functions. In Theorem 6.1 below, we state a maximum principle which was first established by oeper [9], see also Kim McCann [4], Trudinger Wang [33,35], and Villani [38] for other proofs and extensions. For any x Ω + and y, y Ω, letting x c x, y = p and x c x, y = p, i.e. Y x, p = y and Y x, p = y, we say that t Y x, 1 t p + tp is the c -segment connecting y and y with respect to x. oeper s maximum principle states as follows. Needless to say, a dual result can be obtained by exchanging the roles of x and y. Theorem 6.1. et U, Ω +, Ω, c satisfy A A3w and Bw. et x, x Ω + and y, y Ω. Then for any t [, 1], there holds f t := c x, y t c x, y t max f, f 1, where t y t is the c -segment connecting y and y with respect to x.

CONTINUITY AN INJECTIVITY OF OPTIMA MAPS 17 For any x Ω +, we let c u x be as in.3 and u x be the subdifferential of u at x defined by u x := {y R n ; u x u x + y, x x + o x x as x x}. 6.1 We give two corollaries below of Theorem 6.1 which we use in this paper. These results were obtained by oeper [9]. Here again, we mention the related references by Kim McCann [4], Trudinger Wang [33, 35], and Villani [38]. Corollary 6.. et U, Ω +, Ω, c, u satisfy A A3w, Bw, and C. For any x Ω +, there holds u x = x c x, c u x, where u x and c u x are as in.3 and 6.1. In particular, x c x, c u x is convex. Corollary 6.3. et U, Ω +, Ω, c, u satisfy A A3w, Bw, and C. For any y Ω, any local minimum point of the function x u x + c x, y is a global minimum point in Ω +. 6.. On the image and inverse image of boundary points. The assumption B of strong c-convexity of domains is used in this paper in order to apply Theorem 6.4 below, which is due to Figalli Kim McCann [14]. Theorem 6.4. et Ω Ω + and Ω be bounded open subsets of R n. et c and u satisfy A A and C. i If c u Λ 1 n in Ω for some Λ 1 > and Ω is strongly c -convex with respect to Ω, then interior points of Ω cannot be mapped by c u to boundary points of Ω, i.e. c u 1 Ω Ω =. ii If c u Λ n in Ω + for some Λ > and Ω + is strongly c-convex with respect to Ω, then boundary points of Ω + cannot be mapped by c u into interior points of Ω, i.e. c u Ω + Ω =. 6.3. Two results on convex sets. We also use in this paper the following two results on convex sets. The first one is John s Theorem [3]. We use this result in both the Alexandrovtype estimates for lower and upper bounds in Sections 3 and 4. Theorem 6.5. For any compact convex subset K of R n with nonempty interior, there exists a n-dimensional ellipsoid such that E K ne, where ne is the dilation of E with respect to its center of mass. Another result on convex sets, Theorem 6.6 below, is used in the proof of the Alexandrovtype estimates for upper bounds Corollary 4.3. This result was established by Figalli Kim McCann [14]. Theorem 6.6. et K be a convex subset of R n = R n R n. et π be the projection of R n onto R n. et x π K and K = π 1 x K. Then there holds H n K H n π K C n K for some C = C n, n >, where H d is the d-dimensional Haussdorf measure and n is the n-dimensional ebesgue measure. References [1] A. Alexandroff, Smoothness of the convex surface of bounded Gaussian curvature, C. R. oklady Acad. Sci. URSS N.S. 36 194, 195 199. [] J. M. Borwein and A. S. ewis, Convex analysis and nonlinear optimization. Theory and examples, CMS Books in Mathematics, 3, Springer-Verlag, New York,. [3] Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 1991, no. 4, 375 417.

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