The Flatness Theorem for Nonsymmetric Convex Bodies via the Local Theory of Banach Spaces

Let L be a lattice in ℝⁿ and K a convex body disjoint from L. The classical Flatness Theorem asserts that then w(K, L), the L-width of K, does not exceed some bound, depending only on the dimension n; this fact was later found relevant to questions in integer programming. Kannan and Lovász (1988) showed that under the above assumptions w(K, L) ≤ Cn², where C is a universal constant. Banaszczyk (1996) proved that w(K, L) ≤ Cn(1 + log n) if K has a centre of symmetry. In the present paper we show that w(K, L) ≤ Cn^3/2 for an arbitrary 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 arising classes of bodies.