Distortion problem explained

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}} (|y _{1|})/(|y _{2|}) \ge λ (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 and .

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 ℓ^p 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 ℓ^p, 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 ∈ 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 that on ℓ¹ 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 ℓ^p-spaces, 1 < p < ∞, the distortion problem was solved affirmatively by, who showed that ℓ² is arbitrarily distortable, using the first known arbitrarily distortable space constructed by .

See also

• Tsirelson space
• Banach space

References

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