Two comments on Dvoretzky’s sphericity theorem

Abstract

For any two positive integersk, l and anyɛ>0 there exists anN(k, l, ɛ) so that given anyl convex bodiesC1, …,Cl symmetric about the origin inEn withn≧N there exists a subspaceEk so that eachCi intersectsEk, or has a projection intoEk, in a set which is nearly spherical (asphericity <ɛ). The measure of the totality ofEk which intersect a given body inEn in a nearly ellipsoidal set is considered and an affine invariant measure is introduced for that purpose.