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On the Dimension of Almost Hilbertian Subspaces of Quotient Spaces

The question of the dimension of almost Hilbertian subspaces is resolved in [1] where it is shown that every Banach space E of dimension n possesses almost Hilbertian subspaces of dimension c(logn), where c is an absolute constant, and that this estimate is the best possible. When the net is spread wider to include quotient spaces and subspaces of quotient spaces we should expect to find instances of almost Hilbertian spaces of much higher dimension; it comes as no surprise to infer at once, from [1, Theorem 2.9], that E possesses either almost Hilbertian subspaces or almost Hilbertian quotient spaces of dimension at least en. The main theorem of this article (Theorem 5) is a result of this type. The question broached here also receives consideration in [2], where similar results are obtained by different methods. The proofs will follow readily from some results on the volume of convex bodies in IR", of which the oldest is an inequality of Santalo [4]. Volumetric considerations have recently held special significance for Banach-space theorists. In [7] the authors proved the existence of so-called Kashin decompositions for some families of Banach spaces, and they were led to define a certain affine invariant associated with an arbitrary finite-dimensional space which they called the volume ratio. We shall use a theorem relating this quantity to the existence of almost Hilbertian subspaces.