16,424 research outputs found
The equivariant Minkowski problem in Minkowski space
The classical Minkowski problem in Minkowski space asks, for a positive
function on , for a convex set in Minkowski space with
space-like boundary , such that is the
Gauss--Kronecker curvature at the point with normal . Analogously to the
Euclidean case, it is possible to formulate a weak version of this problem:
given a Radon measure on the generalized Minkowski problem
in Minkowski space asks for a convex subset such that the area measure of
is .
In the present paper we look at an equivariant version of the problem: given
a uniform lattice of isometries of , given a
invariant Radon measure , given a isometry group of
Minkowski space, with as linear part, there exists a unique convex set
with area measure , invariant under the action of .
The proof uses a functional which is the covolume associated to every
invariant convex set.
This result translates as a solution of the Minkowski problem in flat space
times with compact hyperbolic Cauchy surface. The uniqueness part, as well as
regularity results, follow from properties of the Monge--Amp\`ere equation. The
existence part can be translated as an existence result for Monge--Amp\`ere
equation.
The regular version was proved by T.~Barbot, F.~B\'eguin and A.~Zeghib for
and by V.~Oliker and U.~Simon for . Our method is
totally different. Moreover, we show that those cases are very specific: in
general, there is no smooth -invariant surface of constant
Gauss-Kronecker curvature equal to
Standard finite elements for the numerical resolution of the elliptic Monge-Ampere equation: Aleksandrov solutions
We prove a convergence result for a natural discretization of the Dirichlet
problem of the elliptic Monge-Ampere equation using finite dimensional spaces
of piecewise polynomial C0 or C1 functions. Standard discretizations of the
type considered in this paper have been previous analyzed in the case the
equation has a smooth solution and numerous numerical evidence of convergence
were given in the case of non smooth solutions. Our convergence result is valid
for non smooth solutions, is given in the setting of Aleksandrov solutions, and
consists in discretizing the equation in a subdomain with the boundary data
used as an approximation of the solution in the remaining part of the domain.
Our result gives a theoretical validation for the use of a non monotone finite
element method for the Monge-Amp\`ere equation
On the differential structure of metric measure spaces and applications
The main goals of this paper are: i) To develop an abstract differential
calculus on metric measure spaces by investigating the duality relations
between differentials and gradients of Sobolev functions. This will be achieved
without calling into play any sort of analysis in charts, our assumptions
being: the metric space is complete and separable and the measure is Borel, non
negative and locally finite. ii) To employ these notions of calculus to
provide, via integration by parts, a general definition of distributional
Laplacian, thus giving a meaning to an expression like , where
is a function and is a measure. iii) To show that on spaces with
Ricci curvature bounded from below and dimension bounded from above, the
Laplacian of the distance function is always a measure and that this measure
has the standard sharp comparison properties. This result requires an
additional assumption on the space, which reduces to strict convexity of the
norm in the case of smooth Finsler structures and is always satisfied on spaces
with linear Laplacian, a situation which is analyzed in detail.Comment: Clarified the dependence on the Sobolev exponent of various
objects built in the paper. Updated bibliography. Corrected typo
Generalized minimizers of convex integral functionals, Bregman distance, Pythagorean identities
Integral functionals based on convex normal integrands are minimized subject
to finitely many moment constraints. The integrands are finite on the positive
and infinite on the negative numbers, strictly convex but not necessarily
differentiable. The minimization is viewed as a primal problem and studied
together with a dual one in the framework of convex duality. The effective
domain of the value function is described by a conic core, a modification of
the earlier concept of convex core. Minimizers and generalized minimizers are
explicitly constructed from solutions of modified dual problems, not assuming
the primal constraint qualification. A generalized Pythagorean identity is
presented using Bregman distance and a correction term for lack of essential
smoothness in integrands. Results are applied to minimization of Bregman
distances. Existence of a generalized dual solution is established whenever the
dual value is finite, assuming the dual constraint qualification. Examples of
`irregular' situations are included, pointing to the limitations of generality
of certain key results
Data depth and floating body
Little known relations of the renown concept of the halfspace depth for
multivariate data with notions from convex and affine geometry are discussed.
Halfspace depth may be regarded as a measure of symmetry for random vectors. As
such, the depth stands as a generalization of a measure of symmetry for convex
sets, well studied in geometry. Under a mild assumption, the upper level sets
of the halfspace depth coincide with the convex floating bodies used in the
definition of the affine surface area for convex bodies in Euclidean spaces.
These connections enable us to partially resolve some persistent open problems
regarding theoretical properties of the depth
Fractal Weyl law for open quantum chaotic maps
We study the semiclassical quantization of Poincar\'e maps arising in
scattering problems with fractal hyperbolic trapped sets. The main application
is the proof of a fractal Weyl upper bound for the number of
resonances/scattering poles in small domains near the real axis. This result
encompasses the case of several convex (hard) obstacles satisfying a no-eclipse
condition.Comment: 69 pages, 7 figure
Regularized Optimal Transport and the Rot Mover's Distance
This paper presents a unified framework for smooth convex regularization of
discrete optimal transport problems. In this context, the regularized optimal
transport turns out to be equivalent to a matrix nearness problem with respect
to Bregman divergences. Our framework thus naturally generalizes a previously
proposed regularization based on the Boltzmann-Shannon entropy related to the
Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We
call the regularized optimal transport distance the rot mover's distance in
reference to the classical earth mover's distance. We develop two generic
schemes that we respectively call the alternate scaling algorithm and the
non-negative alternate scaling algorithm, to compute efficiently the
regularized optimal plans depending on whether the domain of the regularizer
lies within the non-negative orthant or not. These schemes are based on
Dykstra's algorithm with alternate Bregman projections, and further exploit the
Newton-Raphson method when applied to separable divergences. We enhance the
separable case with a sparse extension to deal with high data dimensions. We
also instantiate our proposed framework and discuss the inherent specificities
for well-known regularizers and statistical divergences in the machine learning
and information geometry communities. Finally, we demonstrate the merits of our
methods with experiments using synthetic data to illustrate the effect of
different regularizers and penalties on the solutions, as well as real-world
data for a pattern recognition application to audio scene classification
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