Distortion and stabilized structure in Banach spaces; New geometric phenomena for Banach and Hilbert spaces · 1994-ana-220

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

The problem is unresolved. Property 1 says that $X$ is distortable, while Property 2 is the hereditary, uniform form of bounded distortion: one constant must stabilize every equivalent norm on every infinite-dimensional subspace. No citable construction satisfying both properties, and no theorem ruling out all such spaces, was found. This is stronger than merely saying that one particular space is not known to be arbitrarily distortable.

The strongest established progress is structural:

• Vitali D. Milman and Nicole Tomczak-Jaegermann, “Asymptotic $\ell_p$ Spaces and Bounded Distortions,” Banach Spaces (International Workshop, Merida, 1992), Contemporary Mathematics 144, American Mathematical Society, 1993, pp. 173–195, arXiv:math/9209214. They prove that every Banach space of bounded distortion contains an asymptotic $\ell_p$ subspace for some $1\le p\le\infty$. Thus any example sought here must contain highly regular asymptotic structure, but the result neither constructs nor excludes a distortable example.

• Nicole Tomczak-Jaegermann, “Banach Spaces of Type $p$ Have Arbitrarily Distortable Subspaces,” Geometric and Functional Analysis 6 (1996), no. 6, 1074–1082, https://eudml.org/doc/58254. She proves that a space of bounded distortion contains an unconditional basic sequence; consequently spaces of nontrivial type and hereditarily indecomposable spaces have arbitrarily distortable subspaces. This sharply restricts a possible example but does not settle its existence.

• Edward Odell, Nicole Tomczak-Jaegermann, and Roy Wagner, “Proximity to $\ell_1$ and Distortion in Asymptotic $\ell_1$ Spaces,” Journal of Functional Analysis 150 (1997), no. 1, 101–145, DOI: 10.1006/jfan.1997.3106; arXiv:math/9704214. Definition 2.4 gives the same hereditary uniform notion of $D$-bounded distortion as Property 2 (up to an inessential enlargement of $D$). The paper states that no distortable example was known and focuses the search on asymptotic $\ell_1$ spaces with unconditional bases. Theorem 6.8 forces the stabilized asymptotic constants of any such space to have geometric behavior; Corollary 6.9 supplies a condition implying the presence of an arbitrarily distortable subspace; and Theorem 6.10 shows that an asymptotic $\ell_1$ space of bounded distortion must contain a subspace resembling a Tsirelson-type space. These are necessary conditions, not a resolution.

• Edward Odell and Nicole Tomczak-Jaegermann, “On Certain Equivalent Norms on Tsirelson’s Space,” Illinois Journal of Mathematics 44 (2000), no. 1, 51–71, DOI: 10.1215/ijm/1255984953; arXiv:math/9804057. They prove that a natural sequence of equivalent Tsirelson norms cannot distort any subspace beyond one universal constant. Antonios Manoussakis extends this control to the corresponding modified norms in “A Note on Certain Equivalent Norms on Tsirelson’s Space,” Glasgow Mathematical Journal 46 (2004), no. 2, 379–390, DOI: 10.1017/S0017089504001867. These results cover special renormings only; Property 2 quantifies over every equivalent norm on every subspace.

• Pandelis Dodos, Jordi Lopez-Abad, and Stevo Todorcevic, “Banach Spaces and Ramsey Theory: Some Open Problems,” Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 104 (2010), no. 2, 435–450, DOI: 10.5052/RACSAM.2010.27; arXiv:1006.2668. Problems 11 and 12 again record bounded distortion and the Tsirelson-space candidate as open. Later searches, including literature citing the structural and Tsirelson-specific papers, found no full answer.

Accordingly, the remaining gap is the whole existence dichotomy: no separable infinite-dimensional real Banach space is known to be both distortable and uniformly of bounded distortion in the hereditary sense required here.