Algebroidally Integrable Bodies

Abstract

V. Arnold’s problem 1987–14 from his Problems book asks whether there exist bodies with smooth boundaries in $${{\mathbb {R}}}^N$$ (other than the ellipsoids in odd-dimensional spaces) for which the volume of the segment cut by any hyperplane from the body depends algebraically on the hyperplane. We present a series of very realistic candidates for the role of such bodies, and prove that the corresponding volume functions are at least algebroid, in particular their analytic continuations are finitely valued; to prove their algebraicity it remains to check the condition of finite growth.