Related papers: Alexandrov's Approach to the Minkowski Problem

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…

Author of this article created for the first time the method for finding solutions of the Minkowski problem for closed surfaces in Riemannian space.

Some achievements of the late Andrzej Kossakowski in the field of statistical physics and quantum theory are presented. We recall his attempt to find an analytical solution of the 3-dimensional Ising model and present a problem concerning…

This is a short overview of the influence of mathematicians and their ideas on the creative contribution of Mikhailo Lomonosov on the occasion of the tercentenary of his birth.

The $L_p$-Minkowski problem deals with the existence of closed convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed $p$-area measures. It extends the classical Minkowski problem and embraces several important geometric and physical…

We present a technique for proving convergence to the Aleksandrov solution of the Monge-Ampere equation of a stable and consistent finite difference scheme. We also require a notion of discrete convexity with a stability property and a…

Reminiscences about Alexandr Danilovich Alexandrov (1912--1999)

Appearing in 1921 as an equation for small-amplitude waves on the surface of an infinitely deep liquid, the Nekrasov equation quickly became a source of new results. This manifested itself both in the field of mathematics (theory of…

This paper is devoted to Markov's extremal problems of the form $M_{n,k}=\sup_{p\in\PP_n\setminus\{0\}}{{\|p^{(k)}\|}_X}/{{\|p\|}_X}$ $(1\le k\le n)$, where $\PP_n$ is the set of all algebraic polynomials of degree at most $n$ and $X$ is a…

In this paper, we obtain the existence result of smooth solutions to the Orlicz-Aleksandrov problem from the perspective of geometric flow. Furthermore, a special uniqueness result of solutions to this problem shall be discussed.

In his book on Convex Polyhedra (section 7.2), A.D. Aleksandrov raised a general question of finding variational statements and proofs of existence of polytopes with given geometric data. The first goal of this paper is to give a…

In this paper we provide an extension to the Jellett-Minkowski's formula for immersed submanifolds into ambient manifolds which possesses a pole and radial curvatures bounded from above or below by the radial sectional curvatures of a…

In this paper, we study a class of singular double phase problems defined on Minkowski spaces in terms of Finsler manifolds and with right-hand sides that allow a certain type of critical growth for such problems. Under very general…

This paper is concerned with the regularization of large-scale discrete inverse problems by means of inexact Krylov methods. Specifically, we derive two new inexact Krylov methods that can be efficiently applied to unregularized or…

This article describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors.…

Existence of symmetric solutions to the Gaussian Minkowski problem was established by Huang, Xi and Zhao. In this paper, we show the existence of non-symmetric solutions to this problem by studying the related Monge-Amp\`{e}re type equation…

With the goal of solving optimisation problems on non-Riemannian manifolds, such as geometrical surfaces with sharp edges, we develop and prove the convergence of a forward-backward method in Alexandrov spaces with curvature bounded both…

In this paper, {we extend the affine dual curvature measures to the $L_p$ setting and solve the existence part of the corresponding Minkowski problem for non-symmetric discrete measures when $p>1$ and for symmetric measures when $p\geq0$.}…

This article celebrates the 40th anniversary of Dr. Ivan Dmitrievich Remizov, a mathematician who made a number of important contributions to the theory of one-parameter operator semigroups -- a branch of functional analysis which has…

Alexandrov's Theorem states that every metric with the global topology and local geometry required of a convex polyhedron is in fact the intrinsic metric of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a…