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# Strongly extreme points and approximation properties

Published:2017-12-02

• Trond A. Abrahamsen,
Department of Mathematics, University of Agder, Postboks 422, 4604 Kristiansand, Norway.
• Petr Hájek,
Mathematical Institute, Czech Academy of Science, Żitná 25, 115 67 Praha 1, Czech Republic
• Olav Nygaard,
Department of Mathematics, University of Agder, Postboks 422, 4604 Kristiansand, Norway
• Stanimir L. Troyanski,
Institute of Mathematics and Informatics, Bulgarian Academy of Science, bl.8, acad. G. Bonchev str. 1113 Sofia, Bulgaria
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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.
 Keywords: denting point, strongly extreme point, unconditional compact approximation property
 MSC Classifications: 46B20 - Geometry and structure of normed linear spaces 46B04 - Isometric theory of Banach spaces

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