New examples of locally algebraically integrable bodies

Any compact body in ${\mathbb R}^N$ with smooth boundary defines a two-valued function on the space of affine hyperplanes: the volumes of two parts into which these hyperplanes cut the body. This function is never algebraic if $N$ is even and is very rarely algebraic if $N$ is odd: all known examples (found essentially by Archimedes \cite{arch} for $N=3$) of bodies defining algebraic volume functions are these of ellipsoids. We demonstrate a large new series of {\em locally} algebraically integrable bodies with algebraic boundaries in the spaces of arbitrary dimensions, that is, of bodies such that the corresponding volume functions coincide with algebraic ones at least in some open domains of the space of hyperplanes intersecting the body. Further, we discuss (also following Archimedes) the extension of this problem to non-compact bodies