A short proof of Schoenberg's conjecture on positive definite functions

In 1938, I. J. Schoenberg asked for which positive numbers p is the function exp(- parallel to x parallel to(p)) positive definite, where the norm is taken from one of the spaces l(p)(n), q > 2. The solution of the problem was completed in 1991, by showing that for every p epsilon (0,2], the function exp(- parallel to x parallel to(p)) is not positive definite for the l(q)(n) norms with q > 2 and n greater than or equal to 3. We prove a similar result for a more general class of norms, which contains some Orlicz spaces and q-sums, and, in particular, present a simple proof of the answer to Schoenberg's original question. Some consequences concerning isometric embeddings in L-p spaces for 0 < p less than or equal to 2 are also discussed.