  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.
