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Young-Heon Kim

Publications and source records attributed to Young-Heon Kim.

51 records · Page 3Linked to original sources
arXiv2011

A Generalization of Caffarelli's Contraction Theorem via (reverse) Heat Flow

A theorem of L. Caffarelli implies the existence of a map pushing forward a source Gaussian measure to a target measure which is more log-concave than the source one, which contracts Euclidean distance (in fact, Caffarelli showed that the optimal-transport Brenier map $T_{opt}$ is a contraction in this case). We generalize this result to more general source and target measures, using a condition on the third derivative of the potential, using two different proofs. The first uses a map $T$, whose inverse is constructed as a flow along an advection field associated to an appropriate heat-diffusion process. The contraction property is then reduced to showing that log-concavity is preserved along the corresponding diffusion semi-group, by using a maximum principle for parabolic PDE. In particular, Caffarelli's original result immediately follows by using the Ornstein-Uhlenbeck process and the Prékopa--Leindler Theorem. The second uses the map $T_{opt}$ by generalizing Caffarelli's argument, employing in addition further results of Caffarelli. As applications, we obtain new correlation and isoperimetric inequalities.

math.AP↗
arXiv2011

Hölder continuity and injectivity of optimal maps

Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x,y). If the source density f^+(x) is bounded away from zero and infinity in an open region U' \subset R^n, and the target density f^-(y) is bounded away from zero and infinity on its support \cl V \subset R^n, which is strongly c-convex with respect to U', and the transportation cost c satisfies the A3w condition of Trudinger and Wang \cite{TrudingerWang07p}, we deduce local Hölder continuity and injectivity of the optimal map inside U' (so that the associated potential u belongs to C^{1,α}_{loc}(\U')). Here the exponent α>0 depends only on the dimension and the bounds on the densities, but not on c. Our result provides a crucial step in the low/interior regularity setting: in a sequel \cite{FigalliKimMcCann09p}, we use it to establish regularity of optimal maps with respect to the Riemannian distance squared on arbitrary products of spheres. Three key tools are introduced in the present paper. Namely, we first find a transformation that under A3w makes c-convex functions level-set convex (as was also obtained independently from us by Liu \cite{Liu09}). We then derive new Alexandrov type estimates for the level-set convex c-convex functions, and a topological lemma showing optimal maps do not mix interior with boundary. This topological lemma, which does not require A3w, is needed by Figalli and Loeper \cite{FigalliLoeper08p} to conclude continuity of optimal maps in two dimensions. In higher dimensions, if the densities f^\pm are Hölder continuous, our result permits continuous differentiability of the map inside U' (in fact, C^{2,α}_{loc} regularity of the associated potential) to be deduced from the work of Liu, Trudinger and Wang \cite{LiuTrudingerWang09p}.

math.AP↗
arXiv2011

On supporting hyperplanes to convex bodies

Given a convex set and an interior point close to the boundary, we prove the existence of a supporting hyperplane whose distance to the point is controlled, in a dimensionally quantified way, by the thickness of the convex set in the orthogonal direction. This result has important applications in the regularity theory for Monge-Ampère type equations arising in optimal transportation.

math.AP↗
arXiv2010

Parabolic optimal transport equations on manifolds

We study a parabolic equation for finding solutions to the optimal transport problem on compact Riemannian manifolds with general cost functions. We show that if the cost satisfies the strong MTW condition and the stay-away singularity property, then the solution to the parabolic flow with any appropriate initial condition exists for all time and it converges exponentially to the solution to the optimal transportation problem. Such results hold in particular, on the sphere for the distance squared cost of the round metric and for the far-field reflector antenna cost, among others.

math.DG↗
arXiv2010

Regularity of optimal transport maps on multiple products of spheres

This article addresses regularity of optimal transport maps for cost="squared distance" on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and dimensions. Such manifolds are known to be non-negatively cross-curved [KM2]. Under boundedness and non-vanishing assumptions on the transfered source and target densities we show that optimal maps stay away from the cut-locus (where the cost exhibits singularity), and obtain injectivity and continuity of optimal maps. This together with the result of Liu, Trudinger and Wang [LTW] also implies higher regularity (C^{1,α}/C^\infty) of optimal maps for more smooth (C^α/C^\infty)) densities. These are the first global regularity results which we are aware of concerning optimal maps on non-flat Riemannian manifolds which possess some vanishing sectional curvatures. Moreover, such product manifolds have potential relevance in statistics (see [S]) and in statistical mechanics (where the state of a system consisting of many spins is classically modeled by a point in the phase space obtained by taking many products of spheres). For the proof we apply and extend the method developed in [FKM1], where we showed injectivity and continuity of optimal maps on domains in R^n for smooth non-negatively cross-curved cost. The major obstacle in the present paper is to deal with the non-trivial cut-locus and the presence of flat directions.

math.AP↗
arXiv2010

Pseudo-Riemannian geometry calibrates optimal transportation

Given a transportation cost $c: M \times\bar M \to\mathbf{R}$, optimal maps minimize the total cost of moving masses from $M$ to $\bar M$. We find a pseudo-metric and a calibration form on $M\times\bar M$ such that the graph of an optimal map is a calibrated maximal submanifold. We define the mass of space-like currents in spaces with indefinite metrics.

math.DG↗
arXiv2009

When is multidimensional screening a convex program?

A principal wishes to transact business with a multidimensional distribution of agents whose preferences are known only in the aggregate. Assuming a twist (= generalized Spence-Mirrlees single-crossing) hypothesis and that agents can choose only pure strategies, we identify a structural condition on the preference b(x,y) of agent type x for product type y -- and on the principal's costs c(y) -- which is necessary and sufficient for reducing the profit maximization problem faced by the principal to a convex program. This is a key step toward making the principal's problem theoretically and computationally tractable; in particular, it allows us to derive uniqueness and stability of the principal's optimum strategy -- and similarly of the strategy maximizing the expected welfare of the agents when the principal's profitability is constrained. We call this condition non-negative cross-curvature: it is also (i) necessary and sufficient to guarantee convexity of the set of b-convex functions, (ii) invariant under reparametrization of agent and/or product types by diffeomorphisms, and (iii) a strengthening of Ma, Trudinger and Wang's necessary and sufficient condition (A3w) for continuity of the correspondence between an exogenously prescribed distribution of agents and of products. We derive the persistence of economic effects such as the desirability for a monopoly to establish prices so high they effectively exclude a positive fraction of its potential customers, in nearly the full range of non-negatively cross-curved models.

math.OC↗
arXiv2009

Continuity and injectivity of optimal maps for non-negatively cross-curved costs

Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x,y). If the source density f^+(x) is bounded away from zero and infinity in an open region U' \subset R^n, and the target density f^-(y) is bounded away from zero and infinity on its support clV \subset R^n, which is strongly c-convex with respect to U', and the transportation cost c is non-negatively cross-curved, we deduce continuity and injectivity of the optimal map inside U' (so that the associated potential u belongs to C^1(U')). This result provides a crucial step in the low/interior regularity setting: in a subsequent paper [15], we use it to establish regularity of optimal maps with respect to the Riemannian distance squared on arbitrary products of spheres. The present paper also provides an argument required by Figalli and Loeper to conclude in two dimensions continuity of optimal maps under the weaker (in fact, necessary) hypothesis A3w [17]. In higher dimensions, if the densities f^\pm are Hölder continuous, our result permits continuous differentiability of the map inside U' (in fact, C^{2,α}_{loc} regularity of the associated potential) to be deduced from the work of Liu, Trudinger and Wang [33].

math.AP↗
arXiv2008

Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular)

The key condition A3w of Ma, Trudinger and Wang for regularity of optimal transportation maps is implied by the nonnegativity of a pseudo-Riemannian curvature -- which we call cross-curvature -- induced by the transportation cost. For the Riemannian distance squared cost, it is shown that (1) cross-curvature nonnegativity is preserved for products of two manifolds; (2) both A3w and cross-curvature nonnegativity are inherited by Riemannian submersions; and (3) the $n$-dimensional round sphere satisfies cross-curvature nonnegativity. From these results, a large new class of Riemannian manifolds satisfying cross-curvature nonnegativity (thus A3w) is obtained, including many whose sectional curvature is far from constant. All known obstructions to the regularity of optimal maps are absent from these manifolds, making them a class for which it is natural to conjecture that regularity holds. This conjecture is confirmed for complex projective space CP^n.

math.DG↗
arXiv2007

Continuity, curvature, and the general covariance of optimal transportation

Let M and \bar M be n-dimensional manifolds equipped with suitable Borel probability measures ρand \barρ. Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c \in C^4(M \times \bar M) to guarantee smoothness of the optimal map pushing ρforward to \barρ; the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions to be largely dictated by the covariance of the question; it expresses them via non-negativity of the sectional curvature of certain null-planes in a novel but natural pseudo-Riemannian geometry which the cost c induces on the product space M \times \bar M. Hölder continuity of optimal maps was established for rougher mass distributions by Loeper, still relying on a key result of Trudinger & Wang which required certain structure on the domains and the cost. We go on to develop this theory for mass distributions on differentiable manifolds -- recovering Loeper's Riemannian examples such as the round sphere as particular cases -- give a direct proof of the key result mentioned above, and revise Loeper's Hölder continuity argument to make it logically independent of all earlier works, while extending it to less restricted geometries and cost functions even for subdomains M and \bar M of R^n. We also give new examples of geometries satisfying the hypotheses -- obtained using submersions and tensor products -- and some connections to spacelike Lagrangian submanifolds in symplectic geometry.

math.DG↗
arXiv2007

On the cost-subdifferentials of cost-convex functions

We are interested in the cost-convex potentials in optimal mass transport theory, and we show by direct and geometric arguments the equivalence between cost-subdifferentials and ordinary subdifferentials of cost-convex functions, under the assumptions A0, A1, A2, and A3W on cost functions introduced by Ma, Trudinger, and Wang. The connectivity of contact sets of optimal transport maps follows as a direct corollary. Our approach is quite different from the previous result of Loeper which he obtained as the first step toward his Hoelder regularity theory of potential functions, and which was based upon approximation using the regularity theory of Ma, Trudinger, and Wang. The result in this paper improves his result, by relaxing certain geometrical assumptions on the domains and targets; it also completes his Hoelder regularity theory of potential functions on the round sphere, by making it self-contained.

math.AP↗
arXiv2007

Surfaces with boundary: their uniformizations, determinants of Laplacians, and isospectrality

Let Σbe a compact surface of type (g, n), n > 0, obtained by removing n disjoint disks from a closed surface of genus g. Assuming χ(Σ)<0, we show that on Σ, the set of flat metrics which have the same Laplacian spectrum of Dirichlet boundary condition is compact in the C^\infty topology. This isospectral compactness extends the result of Osgood, Phillips, and Sarnak \cite{OPS3} for type (0,n) surfaces, whose examples include bounded plane domains. Our main ingredients are as following. We first show that the determinant of the Laplacian is a proper function on the moduli space of geodesically bordered hyperbolic metrics on Σ. Secondly, we show that the space of such metrics is homeomorphic (in the C^\infty-topology) to the space of flat metrics (on Σ) with constantly curved boundary. Because of this, we next reduce the complicated degenerations of flat metrics to the simpler and well-known degenerations of hyperbolic metrics, and we show that determinants of Laplacians of flat metrics on Σ, with fixed area and boundary of constant geodesic curvature, give a proper function on the corresponding moduli space. This is interesting because Khuri \cite{Kh} showed that if the boundary length (instead of the area) is fixed, the determinant is not a proper function when Σis of type (g, n), g>0; while Osgood, Phillips, and Sarnak \cite{OPS3} showed the properness when g=0.

math.DG↗
arXiv2005

Holomorphic Extensions of Laplacians and Their Determinants

The Teichmueller space Teich(S) of a surface S in genus g>1 is a totally real submanifold of the quasifuchsian space QF(S). We show that the determinant of the Laplacian det'(Δ) on Teich(S) has a unique holomorphic extension to QF(S). To realize this holomorphic extension as the determinant of differential operators on S, we introduce a holomorphic family {Δ_{μ,ν}} of elliptic second order differential operators on S whose parameter space is the space of pairs of Beltrami differentials on S and which naturally extends the Laplace operators of hyperbolic metrics on S. We study the determinant of this family {Δ_{μ,ν}} and show how this family realizes the holomorphic extension of det'(Δ) as its determinant.

math.CV↗
arXiv2004

Holomorphic extensions of determinants of Laplacians

The Teichmuller space Teich(S) of a surface S in genus g>1 is a real submanifold of the quasifuchsian space QF(S). We show that the determinant of the Laplacian det'(Delta) on Teich(S) has a unique holomorphic extension to QF(S).

math.CV↗