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arXiv · 1910.01626

The isomorphic Kottman constant of a Banach space

Jesús M. F. Castillo·Manuel González·Tomasz Kania·Pier Luigi Papini
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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.

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Jesús M. F. Castillo, Manuel González, Tomasz Kania, Pier Luigi Papini. 2019-10-03. The isomorphic Kottman constant of a Banach space. https://arxiv.org/abs/1910.01626

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