The isomorphic Kottman constant of a Banach space

0531287 - MÚ 2021 RIV US eng J - Journal Article
Castillo, J. M. F. - González, M. - Kania, Tomasz - Papini, P.
The isomorphic Kottman constant of a Banach space.
Proceedings of the American Mathematical Society. Roč. 148, č. 10 (2020), s. 4361-4375. ISSN 0002-9939. E-ISSN 1088-6826
R&D Projects: GA ČR(CZ) GJ19-07129Y
Institutional support: RVO:67985840
Keywords : Kottman constant * Banach space * twisted sum * separated set
OECD category: Pure mathematics
Impact factor: 1.016, year: 2020
Method of publishing: Limited access
https://doi.org/10.1090/proc/15079

We show that the Kottman constant $ K(\cdot )$, together with its symmetric and finite variations, is continuous with respect to the Kadets metric, and they are log-convex, hence continuous, with respect to the interpolation parameter in a complex interpolation schema. Moreover, we show that $ K(X)\cdot K(X^*)\geqslant 2$ for every infinite-dimensional Banach space $ X$.
We also consider the isomorphic Kottman constant (defined as the infimum of the Kottman constants taken over all renormings of the space) and solve the main problem left open in [Banach J. Math. Anal. 11 (2017), pp. 348-362], namely that the isomorphic Kottman constant of a twisted-sum space is the maximum of the constants of the respective summands. Consequently, the Kalton-Peck space may be renormed to have the Kottman constant arbitrarily close to $ \sqrt {2}$. For other classical parameters, such as the Whitley and the James constants, we prove the continuity with respect to the Kadets metric.

Permanent Link: http://hdl.handle.net/11104/0309973