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,

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

(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. Distortability first emerged as an important property of Banach spaces in the 1960s, where it was studied by James (1964) and Milman (1971).

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.

In separable spaces, distortability is equivalent to the ostensibly more general question of whether or not every real-valued Lipschitz function f 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.

In a separable Hilbert space, the distortion problem is equivalent to the question of whether there exist subsets of the unit sphere that are 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 spacs as on other Banach spaces. On a separable Hilbert space, the distortion problem was solved affirmatively by Odell & Schlumprecht (1994).

See also

• Tsirelson space

References

• James, R.C. (1964), "Uniformly nonsquare Banach spaces", Annals of Mathematics 80 (2): 542–550 .
• Milman (1971), "Geometry of Banach spaces II, geometry of the unit sphere", Russian Math Surveys 26: 79–163 .
• Odell, E; Schlumprecht, Th. (2003), "Distortion and asymptotic structure", in Johnson, Lindenstrauss, Hanbook of the geometry of Banach spaces, Volume 2, Elsevier, ISBN 9780444513052 .
• Odell, E.; Schlumprecht, Th. (1993), "The distortion problem of Hilbert space", Geom.Funct.Anal 3: 201–207, doi:10.1007/BF01896023, ISSN 1016-443X, MR1209302 .
• Odell, E.; Schlumprecht, Th. (1994), "The distortion problem", Acta Mathematica 173: 259–281, doi:10.1007/BF02398436, ISSN 0001-5962, MR1301394 .