The Problem With Given Integral Gaussian Curvature

Convex geometry is an important branch of modern geometric analysis,the BrunnMinkowski theory is the core content of convex geometry.Aleksandrov problem,also known as the prescribed integral Gaussian curvature problem,is one of the most essential problems of the Brunn-Minkowski theory.Aleksandrov problem studies the characterization of convex bodies,and mainly involves the existence,uniqueness,continuity and regularity of solution.In this paper,we mainly study the necessary and sufficient conditions for the a-priori C⁰ estimate for the solution of Aleksandrov problem.The Aleksandrov problem posed by Aleksandrov is to find conditions on a given measure μ on Sⁿ under which there exists an F ∈ Fⁿ such that Hⁿ（αF（ω））=μ（ω）for all Borel subset ω（?） Sⁿ.Aleksandrov used his mapping lemma proving first uniqueness among polytopes up to a homothety with respect to O,then Oliker gave a variational solution to the problem of existence and uniqueness of a closed convex hypersurface in the Euclidean space with prescribed integral Gaussian curvature.A-priori C⁰ estimate for Aleksandrov problem are established by Treibergs,and conditions are given which are both necessary and sufficient for the boundness of the hypersphere.For some α∈（0,π/2）,if convex body K satisfies Hⁿ-1（αK（ω））≤H^n-1（ω_α）for all Borel subset ω（?）Sⁿ,where ω_α is the α-neighborhood of ω,then the convex body K has the ratio of radii bounded maxρK/minρK≤c（n,α）<∞.Treibergs gave an example to show how far Aleksandrov’s original condition for the existence of solutions must be strengthened to estimate the bounded.Based on the research results of duality measure,we give a new proof of the a-priori C⁰ estimate for Aleksandrov problem by using blasting analysis and reductio.