Another Low-Technology Estimate in Convex Geometry

Summary

We give a short argument that for some C > 0, every ndimensional Banach ball K admits a 256-round sub quotient 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 ⟼ -x. We define K^o ⊂ V*, the dual of a ball K ⊂ V, by A ball K is the unit ball of a unique Banach norm ||·|| _K defined by 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^o) 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 r -dimensional ball K is r-semiround [8] if it contains an ellipsoid E such that It is r-round if it contains an ellipsoid E such that K ⊆ r E. Santal6's inequality states that if K is an n-dimensional ball and E is an n-dimensional ellipsoid.