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arXiv · 1705.02625

Strongly extreme points and approximation properties

Trond A. Abrahamsen·Petr Hájek·Olav Nygaard·Stanimir Troyanski
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Abstract

We show that if $x$ is a strongly extreme point of a bounded closed convex subset of a Banach space and the identity has a geometrically and topologically good enough local approximation at $x$, then $x$ is already a denting point. It turns out that such an approximation of the identity exists at any strongly extreme point of the unit ball of a Banach space with the unconditional compact approximation property. We also prove that every Banach space with a Schauder basis can be equivalently renormed to satisfy the sufficient conditions mentioned. In contrast to the above results we also construct a non-symmetric norm on $c_0$ for which all points on the unit sphere are strongly extreme, but none of these points are denting.

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Trond A. Abrahamsen, Petr Hájek, Olav Nygaard, Stanimir Troyanski. 2017-05-07. Strongly extreme points and approximation properties. https://doi.org/10.4153/cmb-2017-067-3

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