The flatness theorem for nonsymmetric convex bodies via the local theory of Banach spaces

Let L be a lattice in R-n and Ka convex body disjoint from L. The classical Flatness Theorem asserts that then w(K, L), the L-width of K, does not exc eed some bound, depending only on the dimension n; this fact was later foun d relevant to questions in integer programming. Kannan and Lovasz (1988) sh owed that under the above assumptions w(K, L) less than or equal to Cn(2), where C is a universal constant. Banaszczyk (1996) proved that w(K, L) less than or equal to Cn(1 + log n) if K has a centre of symmetry. In the prese nt paper we show that w(K, L) less than or equal to Cn(3/2) for an arbitrar y K. It is conjectured that the exponent 3/2 may be replaced by 1, perhaps at the cost of a logarithmic factor; we prove that for some naturally arisi ng classes of bodies.