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Moduli of local uniform rotundity for convex bodies in normed spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices General convexity

Published online by Cambridge University Press:  10 April 2025

Carlo Alberto De Bernardi  and
Libor Veselý
Carlo Alberto De Bernardi*
Affiliation:
Dipartimento di Matematica per le Scienze economiche, finanziarie ed attuariali, Università Cattolica del Sacro Cuore, Milano 20123, Italy
Libor Veselý
Affiliation:
Dipartimento di Matematica, Università degli Studi, Milano 20133, Italy e-mail: libor.vesely@unimi.it
*
e-mail: carloalberto.debernardi@unicatt.it carloalberto.debernardi@gmail.com
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Abstract

Let C be a convex body (i.e., a proper closed convex subset with nonempty interior) in a normed space X. We consider four moduli of local uniform rotundity for C at a given point $x\in \partial C$, that extend in a natural way the corresponding notions for the unit ball $B_X$, and we prove that they all coincide. This extends a known result of J. Daneš from 1976 concerning the particular case when $C=B_X$.

Keywords

Normed linear spacemodulus of uniform rotunditymodulus of local uniform rotundityconvex bodypoint of local uniform rotundity

MSC classification

Primary: 52A07: Convex sets in topological vector spaces 46B99: None of the above, but in this section
Secondary: 46B20: Geometry and structure of normed linear spaces 52A99: None of the above, but in this section
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 11
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society Society

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Footnotes

The research of the first author was supported by the INdAM - GNAMPA Project, CUP E53C23001670001, and by MICINN project PID2020-112491GB-I00 (Spain). The research of the second author was supported by the INdAM - GNAMPA Project, CUP E53C23001670001, and by the University of Milan, Research Support Plan PSR 2022.

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