  V. Arnold's problem 1987-14 asks whether there exist smooth hypersurfaces in
$R^N$ (other than the conics in odd-dimensional spaces) for which the volume of
the segment cut by any hyperplane from the body bounded by such a hypersurface
is an algebraic function of the hyperplane. We desribe very realistic
candidates for the role of such new hypersurfaces: in particular, it are
examples (additional to Archimedes' conics) of such hypersurfaces, for which
the analytic continuation of this volume function is finitely valued.
