Doubling condition at the origin for non-negative positive definite functions

TL;DR

This paper investigates the bounds and asymptotic behavior of the sharp constant in a doubling condition at the origin for non-negative positive definite functions, involving convex bodies in Euclidean space.

Contribution

It provides new estimates and asymptotic analysis of the sharp constant in the doubling condition for positive definite functions over convex bodies.

Findings

Derived bounds for the sharp constant C in the doubling condition.

Analyzed the asymptotic behavior of C as the convex bodies vary.

Established relationships between the geometry of U, V and the constant C.

Abstract

We study upper and lower estimates as well as the asymptotic behavior of the sharp constant $C=C_n(U,V)$ in the doubling-type condition at the origin \[ \frac{1}{|V|}\int_{V}f(x)\,dx\le C\,\frac{1}{|U|}\int_{U}f(x)\,dx, \] where $U,V\subset \mathbb{R}^{n}$ are $0$-symmetric convex bodies and $f$ is a non-negative positive definite function.