  Several years ago the authors started looking at some problems of convex
geometry from a more general point of view, replacing volume by an arbitrary
measure. This approach led to new general properties of the Radon transform on
convex bodies including an extension of the Busemann-Petty problem and a
slicing inequality for arbitrary functions. The latter means that the sup-norm
of the Radon transform of any probability density on a convex body of volume
one is bounded from below by a positive constant depending only on the
dimension. In this note, we prove an inequality that serves as an umbrella for
these results
