  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.
