  In 2000, A. Koldobsky asked whether two types of generalizations of the
notion of an intersection-body, are in fact equivalent. The structures of these
two types of generalized intersection-bodies have been studied by the author in
[http://www.arxiv.org/math.MG/0512058], providing substantial positive evidence
for a positive answer to this question. The purpose of this note is to
construct a counter-example, which provides a surprising negative answer to
this question in a strong sense. This implies the existence of non-trivial
non-negative functions in the range of the spherical Radon transform, and the
existence of non-trivial spaces which embed in L_p for certain negative values
of p.
