This is an announcement for the paper "On the extremal rays of the cone of positive, positive definite functions" by Philippe Jaming, Mate Matolcsi, and Szilard Gy. Revesz. 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. Archive classification: math.CA math.FA math.PR Mathematics Subject Classification: 42A82 The source file(s), domain.eps: 12230 bytes The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0801.0941 or http://arXiv.org/abs/0801.0941 or by email in unzipped form by transmitting an empty message with subject line uget 0801.0941 or in gzipped form by using subject line get 0801.0941 to: math@arXiv.org.