The isomorphic Kottman constant of a Banach space

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.