Derivative relationships between volume and surface area of compact regions in R^d

An analytic solution to the Busemann-Petty problem on sections of convex bodies

We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in R^n with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n-1)-dimensional X-ray) gives the ((n-1)-dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in R^n and leads to a unified analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies in R^n such that the ((n-1)-dimensio…

Infinitesimal rigidity of convex surfaces through the second derivative of the Hilbert-Einstein functional I: Polyhedral case

The paper is centered around a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete Hilbert-Einstein functional on the space of "warped polyhedra" with a fixed metric on the boundary. This approach is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra, which deals with deformations that preserve face normals and face areas. In the spherical and in the hyperbolic-de Sitter space, there is a perfect duality between the Hilbert-Einstein functional and the volum…

投影几何中的照明体

Illumination Bodies in Projective Geometries

本文将照明体的概念推广到常曲率黎曼空间和投影芬斯勒几何，证明其体积导数定义了这些背景下凸体的表面积，推广了欧氏空间中的仿射表面积。

内度量与外度量比值及体积

Ratio of Intrinsic Metric to Extrinsic Metric and Volume

本文研究了内度量与外度量比值与面积的关系，并给出了特定曲面比值最大值的下界，同时通过例子说明全局下界不存在。