New Examples of Locally Algebraically Integrable Bodies

Any compact body with regular boundary in ${\mathbb R}^N$ defines a two-valued function on the space of affine hyperplanes: the volumes of the 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 bodies defining algebraic volume functions are ellipsoids (and have been essentially found by Archimedes for $N=3$). We demonstrate a new series of locally algebraically integrable bodies with algebraic boundaries in spaces of arbitrary dimensions, that is, of bodies such that the corresponding volume functions coincide with algebraic ones in some open domains of the space of hyperplanes intersecting the body.