Intrinsic volumes and lattice points of crosspolytopes

Abstract

Hadwiger showed by computing the intrinsic volumes of a regular simplex that a rectangular simplex is a counterexample to Wills' conjecture for the relation between the lattice point enumerator and the intrinsic volumes in dimensions not less than 441. Here we give formulae for the volumes of spherical polytopes related to the intrinsic volumes of the regular crosspolytope and of the rectangular simplex. This completes the determination of intrinsic volumes for regular polytopes. As a consequence we prove that Wills' conjecture is false even for centrally symmetric convex bodies in dimensions not less than 207.