Let φ be a convex function on a strictly convex domain Ω ⊂ Rn, n ≥ 1. The corresponding linearized Monge–Ampère equation is trace(ΦD2u) = f , where Φ := det D2φ (D2φ)−1 is the matrix of cofactors of D2φ. We establish interior Hölder estimates for derivatives of solutions to the equation when the function f on the right hand side belongs to Lp(Ω) for some p > n. The function φ is assumed to be s...

We study the Dirichlet problem for complex Monge–Ampère equations in Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result (Theorem 1.1) extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck in Cn. We also consider the equation on compact manifolds without boundary, attempting to generalize Yau’s theorems in the Kähler case. As applications of the main ...

We study the Monge and Kantorovich transportation problems on R∞ within the class of exchangeable measures. With the help of the de Finetti decomposition theorem the problem is reduced to an unconstrained optimal transportation problem on a Hilbert space. We find sufficient conditions for convergence of finite-dimensional approximations to the Monge solution. The result holds, in particular, un...

We survey some recent developments in the analysis of greedy algorithms for assignment and transportation problems. We focus on the linear programming model for matroids and linear assignment problems with Monge property, on general linear programs, probabilistic analysis for linear assignment and makespan minimization, and on-line algorithms for linear and non-linear assignment problems. Zusam...

We use geometric methods to calculate a formula for the complex Monge-Ampère measure (ddVK) n, for K Rn ⊂ Cn a convex body and VK its Siciak-Zaharjuta extremal function. Bedford and Taylor had computed this for symmetric convex bodies K. We apply this to show that two methods for deriving Bernstein-Markov type inequalities, i.e., pointwise estimates of gradients of polynomials, yield the same r...

We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampère type. These are: the problem of locally prescribed Gaussian curvature for surfaces in R3, and the local isometric embedding problem for two-dimensional Riemannian manifolds. We prove a general local existence result for a large class of degenerate Monge-Ampère...

This paper studies the Vlasov-Monge-Ampère system (VMA), a fully non-linear version of the Vlasov-Poisson system (V P ) where the (real) Monge-Ampère equation det ∂ Ψ ∂xi∂xj = ρ substitutes for the usual Poisson equation. This system can be derived as a geometric approximation of the Euler equations of incompressible fluid mechanics in the spirit of Arnold and Ebin. Global existence of weak sol...

© 2014, Springer-Verlag Berlin Heidelberg. In this paper we study local properties of cost and potential functions in optimal transportation. We prove that in a proper normalization process, the cost function is uniformly smooth and converges locally smoothly to a quadratic cost x · y, while the potential function converges to a quadratic function. As applications we obtain the interior W2, p e...