Related papers: Extremal Alexandrov estimates: singularities, obst…
We present a numerical method for solving the Monge-Ampere equation based on the characterization of the solution of the Dirichlet problem as the minimizer of a convex functional of the gradient and under convexity and nonlinear…
This paper analyzes a regularization scheme of the Monge--Amp\`ere equation by uniformly elliptic Hamilton--Jacobi--Bellman equations. The main tools are stability estimates in the $L^\infty$ norm from the theory of viscosity solutions…
The extremal dependence structure of a regularly varying $d$-dimensional random vector can be described by its angular measure. The standard nonparametric estimator of this measure is the empirical measure of the observed angles of the $k$…
We consider the Monge-Kantorovich optimal transportation problem between two measures, one of which is a weighted sum of Diracs. This problem is traditionally solved using expensive geometric methods. It can also be reformulated as an…
We investigate the properties of convex functions in the plane that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Ampere equation for a Monge-Kantorovich problem with quadratic cost between non-absolutely…
Extreme-value copulas arise in the asymptotic theory for componentwise maxima of independent random samples. An extreme-value copula is determined by its Pickands dependence function, which is a function on the unit simplex subject to…
The angular measure on the unit sphere characterizes the first-order dependence structure of the components of a random vector in extreme regions and is defined in terms of standardized margins. Its statistical recovery is an important step…
A compactness theorem for volume-constrained almost-critical points of elliptic integrands is proven. The result is new even for the area functional, as almost-criticality is measured in an integral rather than in a uniform sense. Two main…
In the Euclidean setting, the well-known Alexandrov theorem states that convex functions are twice differentiable almost everywhere. In this note, we extend this theorem to rank-one convex functions. Our approach is novel in that it draws…
We develop discrete $W^2_p$-norm error estimates for the Oliker-Prussner method applied to the Monge-Amp\`ere equation. This is obtained by extending discrete Alexandroff estimates and showing that the contact set of a nodal function…
The extremal dependence structure of a regularly varying random vector Xis fully described by its limiting spectral measure. In this paper, we investigate how torecover characteristics of the measure, such as extremal coefficients, from the…
We introduce an integral representation of the Monge-Amp\`ere equation, which leads to a new finite difference method based upon numerical quadrature. The resulting scheme is monotone and fits immediately into existing convergence proofs…
In this paper we prove that a strictly convex Alexandrov solution u of the Monge-Amp\`ere equation, with right hand side bounded away from zero and infinity, is $W_{\rm loc}^{2,1}$. This is obtained by showing higher integrability a-priori…
Through the study of the degenerate complex Monge-Amp\`ere equation, we establish the optimal regularity of the extremal function associated to intrinsic norms of Chern-Levine-Nirenberg and Bedford-Taylor. We prove a conjecture of…
We use geometric methods to calculate a formula for the complex Monge-Amp\`ere measure $(dd^cV_K)^n$, for $K \Subset \RR^n \subset \CC^n$ a convex body and $V_K$ its Siciak-Zaharjuta extremal function. Bedford and Taylor had computed this…
To a mesh function we associate the natural analogue of the Monge-Ampere measure. The latter is shown to be equivalent to the Monge-Ampere measure of the convex envelope. We prove that the uniform convergence to a bounded convex function of…
We derive a number of sharp upper bounds for the deficit in the Alexandrov-Fenchel inequality using a weighted Minkowski integral formula and an integral formula for the deficit in Jensen's inequality. Our estimates yield results under…
Let $n\ge1$ and $B\ge2$. A real-valued function $f$ defined on the $n$-simplex $\Delta_n$ is approximately convex with respect to $\Delta_{B-1}$ iff f(\sum_{i=1}^B t_ix_i) \le \sum_{i=1}^B t_if(x_i) +1 for all $x_1,...,x_B \in \Delta_n$ and…
We study the exact effect of the anisotropic convexity of domains on the boundary estimate for two Monge-Amp\`ere Equations: one is singular which is from the proper affine hyperspheres with constant mean curvature; the other is degenerate…
We establish two new estimates which control a function (after subtracting its average) in $L^1$ by only the $L^1$ norm of its radial derivative. While the interior estimate holds for all superharmonic functions, the boundary version is…