The isomorphic Kottman constant of a Banach space

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 [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.