A solution of the finite-dimensional homogeneous Banach space problem

Abstract

It is proved that if a finite-dimensional Banach space X has the property that all its αn-dimensional subspaces are K-isomorphic, for some 0 < α < 1 and K ≥ 1, then X is f(α, K)-isomorphic to a Hilbert space, where f(α, K) is cK3/2, if 0 < α < 2/3 and cK2, if 2/3 < α < 1, and where c = c(α) depends on α only.