Another Low-technology Estimate in Convex Geometry

We give a short argument that for some C > 0, every n-dimensional Banach ball K admits a 256-round subquotient of dimension at least Cn/(logn). This is a weak version of Milman’s quotient of subspace theorem, which lacks the logarithmic factor. Let V be a finite-dimensional vector space over R and let V ∗ denote the dual vector space. A symmetric convex body or (Banach) ball is a compact convex set with nonempty interior which is invariant under under x 7→ −x. We define K ⊂ V , the dual of a ball K ⊂ V , by K = {y ∈ V ∗ ∣ y(K) ⊂ [−1, 1]}. A ball K is the unit ball of a unique Banach norm || · ||K defined by ||v||K = min{t ∣ v ∈ tK}. A ball K is an ellipsoid if || · ||K is an inner-product norm. Note that all ellipsoids are equivalent under the action of GL(V ). If V is not given with a volume form, then a volume such as Vol K for K ⊂ V is undefined. However, some expressions such as (Vol K)(Vol K) or (Vol K)/(Vol K ) for K,K ′ ⊂ V are well-defined, because they are independent of the choice of a volume form on V , or equivalently because they are invariant under GL(V ) if a volume form is chosen. An n-dimensional ball K is r-semiround [8] if it contains an ellipsoid E such that (Vol K)/(Vol E) ≤ r. It is r-round if it contains an ellipsoid E such that K ⊆ rE. Santaló’s inequality states that if K is an n-dimensional ball and E is an n-dimensional ellipsoid, (Vol K)(Vol K) ≤ (Vol E)(Vol E). (Saint-Raymond [7], Ball [1], and Meyer and Pajor [4] have given elementary proofs of Santaló’s inequality.) It follows that if K is r-round, then either K or K is √ r-semiround. If K is a ball in a vector space V and W is a subspace, we define W ∩K to be a slice of K and the image of K in V/W to be a projection of K; they are both balls. Following Milman [5], we define a subquotient of K to be a slice of a projection of K. Note that a slice of a projection is also a projection of a slice, so that we could also have called a subquotient a proslice. It follows that a subquotient of a subquotient is a subquotient. Note also that a slice of K is dual to a projection of K, and therefore a subquotient of K is dual to a proslice (or a subquotient) of K. In this paper we prove the following theorem: Theorem 1. Suppose that K is a (2n)-dimensional ball which is (2 k · 4)-semiround, with k ≥ 0. Then K has a 256-round, n-dimensional subquotient. Corollary 2. There exists a constant C > 0 such that every n-dimensional ball K admits a 256-round subquotient of dimension at least Cn/(logn). The corollary follows from the theorem of John that every n-dimensional ball is ( √ n)-round. The corollary is a weak version of a celebrated result of Milman [5, 6]: Theorem 3 (Milman). For every C > 1, there exists D > 0, and for every D < 1 there exists a C, such that every n-dimensional ball K admits a C-round subquotient of dimension at least Dn. Date: December 4, 1994. 1991 Mathematics Subject Classification. Primary 52A21; Secondary 46B03. The author was supported by an NSF Postdoctoral Fellowship, grant #DMS-9107908. 1