Distortion problem

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title: "Distortion problem" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["functional-analysis"] topic_path: "general/functional-analysis" source: "https://en.wikipedia.org/wiki/Distortion_problem" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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 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 ℓp for some {{nowrap|1 ≤ p p, all of which are separable and uniform convex, for {{nowrap|1

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)| 1 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 ℓp-spaces, 1 2 is arbitrarily distortable, using the first known arbitrarily distortable space constructed by .

References

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  • {{citation |chapter=Distortion and asymptotic structure |first1=E. |last1=Odell |author-link1=Edward Odell |first2=Th. |last2=Schlumprecht |title=Handbook of the geometry of Banach spaces, Volume 2 |editor=Johnson |editor2=Lindenstrauss |publisher=Elsevier |year=2003 |isbn=978-0-444-51305-2}}.
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functional-analysis