Distortion problem

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,

$\sup _{y_{1},y_{2}\in Y,\|y_{i}\|=1}{\frac {|y_{1}|}{|y_{2}|}}\geq \lambda$

(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 ℓ^p 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 ℓ^p, all of which are separable and uniform convex, for 1 < p < ∞.

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 ℓ^p-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).

See also

• Tsirelson space
• Banach space

References

• James, R.C. (1964), "Uniformly nonsquare Banach spaces", Annals of Mathematics, 80 (2): 542–550, doi:10.2307/1970663.
• Milman, V.D. (1971), "Geometry of Banach spaces II, geometry of the unit sphere", Russian Mathematical Surveys, 26: 79–163, Bibcode:1971RuMaS..26...79M, doi:10.1070/RM1971v026n06ABEH001273.
• Odell, E.; Schlumprecht, Th. (2003), "Distortion and asymptotic structure", in Johnson; Lindenstrauss (eds.), Handbook of the geometry of Banach spaces, Volume 2, Elsevier, ISBN 978-0-444-51305-2.
• Odell, E.; Schlumprecht, Th. (1993), "The distortion problem of Hilbert space", Geometric and Functional Analysis, 3: 201–207, doi:10.1007/BF01896023, ISSN 1016-443X, MR 1209302.
• Odell, E.; Schlumprecht, Th. (1994), "The distortion problem", Acta Mathematica, 173: 259–281, doi:10.1007/BF02398436, ISSN 0001-5962, MR 1301394.
• Schlumprecht, Th. (1991), "An arbitrary distortable Banach space", Israel Journal of Mathematics, 76: 81–95, arXiv:math/9201225, doi:10.1007/bf02782845, ISSN 0021-2172, MR 1177333.