In functional analysis, a branch of mathematics, the distortion problem is to determine by how much one can distort the unit sphere in a given Banach space using an equivalent norm. Specifically, a Banach space X is called û-distortable if there exists an equivalent norm |x| on X such that, for all infinite-dimensional subspaces Y in X,
(see distortion (mathematics)). Note that every Banach space is trivially 1-distortable. A Banach space is called distortable if it is û-distortable for some û > 1 and it is called arbitrarily distortable if it is û-distortable for any û. Distortability first emerged as an important property of Banach spaces in the 1960s, where it was studied by and .
James proved that c<sub>0</sub> and âÂÂ<sup>1</sup> are not distortable. Milman showed that if X is a Banach space that does not contain an isomorphic copy of c<sub>0</sub> or âÂÂ<sup>p</sup> for some (see sequence space), then some infinite-dimensional subspace of X is distortable. So the distortion problem is now primarily of interest on the spaces âÂÂ<sup>p</sup>, all of which are separable and uniform convex, for .
In separable and uniform convex spaces, distortability is easily seen to be equivalent to the ostensibly more general question of whether or not every real-valued Lipschitz function ƒ defined on the sphere in X stabilizes on the sphere of an infinite dimensional subspace, i.e., whether there is a real number a â R so that for every ô > 0 there is an infinite dimensional subspace Y of X, so that |a − ƒ(y)| < ô, for all y â Y, with ||y|| = 1. But it follows from the result of that on âÂÂ<sup>1</sup> there are Lipschitz functions which do not stabilize, although this space is not distortable by . In a separable Hilbert space, the distortion problem is equivalent to the question of whether there exist subsets of the unit sphere separated by a positive distance and yet intersect every infinite-dimensional closed subspace. Unlike many properties of Banach spaces, the distortion problem seems to be as difficult on Hilbert spaces as on other Banach spaces. On a separable Hilbert space, and for the other âÂÂ<sup>p</sup>-spaces, 1 < p < âÂÂ, the distortion problem was solved affirmatively by , who showed that âÂÂ<sup>2</sup> is arbitrarily distortable, using the first known arbitrarily distortable space constructed by .