Convex bodies with few faces

Abstract

It is proved that if u, u,n are vectors in R, k r. An application to number theory is stated. 0. INTRODUCTION In [V], Vaaler proved that if Q, = [v I]n is the central unit cube in Rn and U is a subspace of Rn then the volume I U n Qn , of the section of Qn by U is at least 1. This result may be reformulated as follows: if u I I... Un are vectors in Rk, 1 2. A related theorem, (Theorem 1, below) in which the condition E u, 2 O so that if u, ...,uE R 1 /1 The estimate is best possible if n is at most exponential in k, apart from the value of the constant J. This is demonstrated by an example which had Received by the editors September 14, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 52A20, 1OE05. The first author was partially supported by N. S. F. DMS-8807243. ? 1990 Americani Mathematical Society 0002-9939/90 $1.00 + $.25 per page