We study the Banach spaces X with the following property: there is a number in ]0,1[ such that for some constant C, any finite dimensional subspace E X contains a subspace F E with dim F dim E which is C-isomorphic to a Euclidean space. We show that if this holds for some in ]0,1[ then it also holds for all in ]0,1[ and we estimate the function C=C(). We show that this property holds iff the "volume ratio" of the finite dimensional subspaces of X are uniformly bounded. We also show that (although X can have this property without being of cotype 2)L 2(X) possesses this property iff X if of cotype 2. In the last part of the paper, we study the K-convex spaces which have a dual with the above property and we relate it to a certain extension property. 1986 Hebrew University.