The core of the unit sphere of a Banach space

Almudena Campos-Jiménez; Francisco Javier García-Pacheco; Almudena Campos-Jiménez; Francisco Javier García-Pacheco

doi:10.3934/math.2024169

AIMS Mathematics

2024, Volume 9, Issue 2: 3440-3452. doi: 10.3934/math.2024169

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The core of the unit sphere of a Banach space

Almudena Campos-Jiménez ¹,
Francisco Javier García-Pacheco ^{2
,
,}

1.
Department of Algebra, Geometry and Topology, Faculty of Sciences, University of Malaga, Campus de Teatinos, Málaga 29071, Spain
2.
Department of Mathematics, College of Engineering, University of Cadiz, Avda. de la Universidad 10, Puerto Real 11519, Spain

Received: 23 October 2023 Revised: 18 December 2023 Accepted: 02 January 2024 Published: 08 January 2024
MSC : 46B20

A geometric invariant or preserver is essentially a geometric property of the unit sphere of a real Banach space that remains invariant under the action of a surjective isometry onto the unit sphere of another real Banach space. A new geometric invariant of the unit ball of a real Banach space was introduced and analyzed in this manuscript: The core of the unit sphere. This geometric invariant consists of all points in the unit sphere of a real Banach space, which are contained in a unique maximal face. It is, in a geometrical sense, the opposite of fractal-like sets such as starlike sets. Classical geometric properties, such as smoothness and strict convexity, were employed to characterize the core of the unit sphere. Also, the core was related to a recently introduced new index: the index of strong rotundity. A characterization of the core in terms of the index of strong rotundity was provided. Finally, applications to longstanding open problems, such as Tingley's problem, were provided by presenting a new notion: Mazur-Ulam classes of Banach spaces.
- Tingley's Problem,
- Mazur-Ulam property,
- surjective isometry,
- geometric invariant,
- extreme point,
- exposed point,
- face,
- facet,
- strict convexity, smoothness
Citation: Almudena Campos-Jiménez, Francisco Javier García-Pacheco. The core of the unit sphere of a Banach space[J]. AIMS Mathematics, 2024, 9(2): 3440-3452. doi: 10.3934/math.2024169

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Abstract

A geometric invariant or preserver is essentially a geometric property of the unit sphere of a real Banach space that remains invariant under the action of a surjective isometry onto the unit sphere of another real Banach space. A new geometric invariant of the unit ball of a real Banach space was introduced and analyzed in this manuscript: The core of the unit sphere. This geometric invariant consists of all points in the unit sphere of a real Banach space, which are contained in a unique maximal face. It is, in a geometrical sense, the opposite of fractal-like sets such as starlike sets. Classical geometric properties, such as smoothness and strict convexity, were employed to characterize the core of the unit sphere. Also, the core was related to a recently introduced new index: the index of strong rotundity. A characterization of the core in terms of the index of strong rotundity was provided. Finally, applications to longstanding open problems, such as Tingley's problem, were provided by presenting a new notion: Mazur-Ulam classes of Banach spaces.

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