Related papers: The Discrete Gauss Image Problem
We survey some results on scalar curvature and properties of solutions to the Einstein constraint equations. Topics include an extended discussion of asymptotically flat solutions to the constraint equations, including recent results on the…
The present work has a double aim. On the one hand we call attention on the relationship existing between the Ashtekar formalism and other gauge-theoretical approaches to gravity, in particular the Poincar\'e Gauge Theory. On the other hand…
We propose three iterative methods for solving the Moser-Veselov equation, which arises in the discretization of the Euler-Arnold differential equations governing the motion of a generalized rigid body. We start by formulating the problem…
In this paper we study the problem, posed by Troyanov, of prescribing the Gaussian curvature under a conformal change of the metric on surfaces with conical singularities. Such geometrical problem can be reduced to the solvability of a…
This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints…
The classical Busemann-Petty problem asks whether smaller central hyperplane sections of origin-symmetric convex bodies necessarily imply smaller total volume. Zvavitch studied this question for arbitrary measures with continuous even…
Given a set $P$ of $n$ points in the plane, its unit-disk graph $G(P)$ is a graph with $P$ as its vertex set such that two points of $P$ are connected by an edge if their (Euclidean) distance is at most $1$. We consider several classical…
The $(2k)$-th Gauss-Bonnet curvature is a generalization to higher dimensions of the $(2k)$-dimensional Gauss-Bonnet integrand, it coincides with the usual scalar curvature for $k=1$. The Gauss-Bonnet curvatures are used in theoretical…
We survey problems and results from combinatorial geometry in normed spaces, concentrating on problems that involve distances. These include various properties of unit-distance graphs, minimum-distance graphs, diameter graphs, as well as…
Given a real number $q$ and a star body in the $n$-dimensional Euclidean space, the generalized dual curvature measure of a convex body was introduced by Lutwak-Yang-Zhang [43]. The corresponding generalized dual Minkowski problem is…
A classical problem in general relativity is the Cauchy problem for the linearised Einstein equation (the initial value problem for gravitational waves) on a globally hyperbolic vacuum spacetime. A well-known result is that it is uniquely…
This paper presents a unified geometric framework for the statistical analysis of a general ill-posed linear inverse model which includes as special cases noisy compressed sensing, sign vector recovery, trace regression, orthogonal matrix…
Gravity inversion allows us to constrain the interior mass distribution of a planetary body using the observed shape, rotation, and gravity. Traditionally, techniques developed for gravity inversion can be divided into Monte Carlo methods,…
Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the…
In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was…
A central conjecture in inverse Galois theory, proposed by D\`{e}bes and Deschamps, asserts that every finite split embedding problem over an arbitrary field can be regularly solved. We give an unconditional proof of a consequence of this…
Gersho's conjecture in 3D asserts the asymptotic periodicity and structure of the optimal centroidal Voronoi tessellation. This relatively simple crystallization problem remains to date open. We prove bounds on the geometric complexity of…
In this paper, the $q$-th dual curvature measure is extended to convex functions and the associated Minkowski problem is posed. A special case includes the $q$-th dual curvature measure of convex bodies which defined by Huang, Lutwak, Yang…
We study the general theorem about gravitational lensing which states the relationship between the numbers of images with different parities. Our formulation allows an extension to the nontransparent and singular model.
We develop a novel framework to study smooth and strongly convex optimization algorithms, both deterministic and stochastic. Focusing on quadratic functions we are able to examine optimization algorithms as a recursive application of linear…