On the extremal rays of the cone of positive, positive definite functions

TL;DR

This paper explores the structure of the cone of non-negative, radial, positive-definite functions, revealing many extremals beyond Gaussians and unifying various characterizations through Choquet integral representations.

Contribution

It characterizes a broad class of extremals of the cone of positive-definite functions, disproving a conjecture that Gaussians are the only extremals, and unifies existing characterizations.

Findings

Many extremals are not Gaussians.

Disproved G. Choquet's conjecture on extremals.

Unified various characterizations via Choquet integrals.

Abstract

The aim of this paper is to investigate the cone of non-negative, radial, positive-definite functions in the set of continuous functions on $\R^d$. Elements of this cone admit a Choquet integral representation in terms of the extremals. The main feature of this article is to characterize some large classes of such extremals. In particular, we show that there many other extremals than the gaussians, thus disproving a conjecture of G. Choquet and that no reasonable conjecture can be made on the full set of extremals. The last feature of this article is to show that many characterizations of positive definite functions available in the literature are actually particular cases of the Choquet integral representations we obtain.