Functions with isotropic sections

Differentiability Properties of Isotropic Functions

1. Introduction. Let Sym denote the linear space of all symmetric second-order tensors on an n-dimensional real vector space Vect with scalar product. (If Vect is identified with R n , then Sym may be identified with the set of all symmetric n-by-n matrices.) A function f : Sym → R is said to be isotropic if f (A) = f (QAQ T) for all A ∈ Sym and all Q proper orthogonals. An isotropic function h...

High Dimensional Random Sections of Isotropic Convex Bodies

We study two properties of random high dimensional sections of convex bodies. In the first part of the paper we estimate the central section function |K ∩F⊥| n−k for random F ∈ Gn,k and K ⊂ R n a centrally symmetric isotropic convex body. This partially answers a question raised by V. Milman and A. Pajor (see [MP], p.88). In the second part we show that every symmetric convex body has random hi...

Representations of certain isotropic tensor functions

The Dagum family of isotropic correlation functions

A function ρ : [0,∞)→ (0,1] is a completely monotonic function if and only if ρ(‖x‖) is positive definite on R for all d and thus it represents the correlation function of a weakly stationary and isotropic Gaussian random field. Radial positive definite functions are also of importance as they represent characteristic functions of spherically symmetric probability distributions. In this paper, ...

From Holomorphic Functions to Holomorphic Sections

It is a pleasure to have the opportunity in the graduate colloquium to introduce my research field. I am a differential geometer. To be more precise, I am a complex differential geometer, although I am equally interested in real differential geometry. To many people, geometry is a kind of mathematics that is related to length, area, volume, etc. For the Euclidean geometry, this is indeed the ca...