Volumetric invariants and operators on random families of Banach spaces

Abstract

The geometry of random projections of centrally symmetric convex bodies inR N is studied. It is shown that if for such a bodyK the Euclidean ballB N 2 is the ellipsoid of minimal volume containing it and a random n-dimensional projection B = PH (K) is \far from PH (B N ) then the (random) body B is as \rigid as its \distance to PH (B N ) permits. The result holds for the full range of dimensions 1 n N , for arbitrary 2 (0; 1). 0. Introduction. The systematic study of random quotients of l n 1 's begun with the acclaimed solution by E. D. Gluskin of the problem of di- ameters of Minkowski compacta (G1). Soon after, several papers followed in which optimal estimates of basic constants or symmetry constants for such quotients were given (S1), (G2), (M1). It was J. Bourgain who rst used this line of argument in the context of general nite-dimensional Banach spaces, and this approach was further developed by the authors in a series of papers (cf. e.g. (MT1), (MT2)). We refer the reader to the survey (MT3) for more information on the subject. The present paper is a further development of the study, begun in (MT4), of geometric and linear properties of families of \random projections of symmetric convex bodies. Here we investigate symmetry properties of such random projections in terms of the existence of well bounded operators on associated Banach spaces, and we relate these properties to average geomet- ric distance of the projections to the Euclidean ball.