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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 theextremals. The main feature of this article is to characterize some largeclasses of such extremals. In particular, we show that there many otherextremals than the gaussians, thus disproving a conjecture of G. Choquet andthat no reasonable conjecture can be made on the full set of extremals. Thelast feature of this article is to show that many characterizations of positivedefinite functions available in the literature are actually particular cases ofthe Choquet integral representations we obtain.

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