Distortion problem

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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 λ-distortable if there exists an equivalent norm |x| on X such that, for all infinite-dimensional subspaces Y in X,

$\sup_{y_1,y_2\in Y, \|y_i\|=1} \frac{|y_1|}{|y_2|} \ge \lambda$

(see distortion (mathematics)). Note that every Banach space is trivially 1-distortable. A Banach space is called distortable if it is λ-distortable for some λ > 1 and it is called arbitrarily distortable if it is λ-distortable for any λ. Distortability first emerged as an important property of Banach spaces in the 1960s, where it was studied by James (1964) and Milman (1971).

James proved that c0 and ℓ1 are not distortable. Milman showed that if X is a Banach space that does not contain an isomorphic copy of c0 or ℓp for some 1 ≤ p < ∞ (see sequence space), then some infinite-dimensional subspace of X is distortable. So the distortion problem is now primarily of interest on the spaces ℓp, all of which are separable and uniform convex.

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 ∈ R so that for every δ > 0 there is an infinite dimensional subspace Y of X, so that |a − ƒ(y)| < δ, for all y ∈ Y, with ||y|| = 1. But it follows from the result of Odell & Schlumprecht (1994) that on ℓ1 there are Lipschitz functions which do not stabilize, although this space is not distortable by James (1964). 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 ℓp-spaces, 1 < p < ∞, the distortion problem was solved affirmatively by Odell & Schlumprecht (1994), who showed that ℓ2 is arbitrarily distortable, using the first known arbitrarily distortable space constructed by Schlumprecht (1991).