In 1938, I. J. Schoenberg asked for which positive numbers p is the function
exp(−∥x∥p) positive
definite, where the norm is taken from one of the spaces
[lscr ]np, q>2. The solution of the
problem was completed in 1991, by showing that for every p∈(0, 2],
the function exp(−∥x∥p) is not positive definite for
the [lscr ]nq norms with q>2
and n[ges ]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 Lp spaces for
0<p[les ]2 are also discussed.