a1 DPMMS, Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WB, England. E-mail: O.Pikhurko@dpmms.cam.ac.uk
It is shown that, for any lattice polytope Pd the set int (P)ld (provided that it is non-empty) contains a point whose coefficient of asymmetry with respect to P is at most 8d · (8l+7)22d+1. If, moreover, P is a simplex, then this bound can be improved to 8 · (8l+7 )2d+1. As an application, new upper bounds on the volume of a lattice polytope are deduced, given its dimension and the number of sublattice points in its interior.
(Received October 06 2000)