The isomorphic Kottman constant of a Banach space

TL;DR

This paper investigates the properties of the Kottman constant in Banach spaces, establishing its continuity, convexity, and behavior under renormings, and solves an open problem regarding its value in twisted-sum spaces.

Contribution

It proves the continuity and log-convexity of the Kottman constant, and solves the open problem about its behavior in twisted-sum Banach spaces.

Findings

Kottman constant is continuous with respect to the Kadets metric.

The isomorphic Kottman constant of a twisted-sum space equals the maximum of its summands' constants.

The Kalton–Peck space can be renormed to have Kottman's constant arbitrarily close to √2.

Abstract

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 [CaGoPa17], 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 Kottman's 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.