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A note on extreme points of $ C^\infty$-smooth balls in polyhedral spaces


Authors: A. J. Guirao, V. Montesinos and V. Zizler
Journal: Proc. Amer. Math. Soc. 143 (2015), 3413-3420
MSC (2010): Primary 46B20; Secondary 46B03, 46B10, 46B22
DOI: https://doi.org/10.1090/S0002-9939-2015-12617-2
Published electronically: April 2, 2015
MathSciNet review: 3348784
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Abstract: Morris (1983) proved that every separable Banach space $ X$ that contains an isomorphic copy of $ c_0$ has an equivalent strictly convex norm such that all points of its unit sphere $ S_X$ are unpreserved extreme, i.e., they are no longer extreme points of $ B_{X^{**}}$. We use a result of Hájek (1995) to prove that any separable infinite-dimensional polyhedral Banach space has an equivalent $ C^{\infty }$-smooth and strictly convex norm with the same property as in Morris' result. We additionally show that no point on the sphere of a $ C^2$-smooth equivalent norm on a polyhedral infinite-dimensional space can be strongly extreme, i.e., there is no point $ x$ on the sphere for which a sequence $ (h_n)$ in $ X$ with $ \Vert h_n\Vert\not \to 0$ exists such that $ \Vert x\pm h_n\Vert\to 1$.


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Additional Information

A. J. Guirao
Affiliation: Instituto de Matemática Pura y Aplicada. Universitat Politècnica de València, C/ Vera, s/n, 46020 Valencia, Spain
Email: anguisa2@mat.upv.es

V. Montesinos
Affiliation: Instituto de Matemática Pura y Aplicada. Universitat Politècnica de València, C/ Vera, s/n, 46020 Valencia, Spain
Email: vmontesinos@mat.upv.es

V. Zizler
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada
Email: vasekzizler@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2015-12617-2
Keywords: Polyhedral space, extreme point, norm that locally depends on a finite number of coordinates, countable James boundary.
Received by editor(s): July 8, 2013
Published electronically: April 2, 2015
Additional Notes: The first author’s research was supported by Ministerio de Economía y Competitividad and FEDER under project MTM2011-25377 and the Universitat Politècnica de València.
The second author’s research was supported by Ministerio de Economía y Competitividad and FEDER under project MTM2011-22417 and the Universitat Politècnica de València.
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2015 American Mathematical Society