RT journal article
T1 Geometric Invariants of Surjective Isometries between Unit Spheres
A1 Campos Jiménez, Almudena
A1 García Pacheco, Francisco Javier
A2 Matemáticas
K1 tingley problem
K1 Mazur-Ulam property
K1 surjective isometry
K1 extension isometries
K1 geometric invariants
K1 extreme point
K1 exposed point
K1 face
K1 facet
K1 strictly convex
AB In this paper we provide new geometric invariants of surjective isometries between unit spheres of Banach spaces. Let X,Y be Banach spaces and let T:SX -> SY be a surjective isometry. The most relevant geometric invariants under surjective isometries such as T are known to be the starlike sets, the maximal faces of the unit ball, and the antipodal points (in the finite-dimensional case). Here, new geometric invariants are found, such as almost flat sets, flat sets, starlike compatible sets, and starlike generated sets. Also, in this work, it is proved that if F is a maximal face of the unit ball containing inner points, then T(-F)=-T(F). We also show that if [x,y] is a non-trivial segment contained in the unit sphere such that T([x,y]) is convex, then T is affine on [x,y]. As a consequence, T is affine on every segment that is a maximal face. On the other hand, we introduce a new geometric property called property P, which states that every face of the unit ball is the intersection of all maximal faces containing it. This property has turned out to be, in a implicit way, a very useful tool to show that many Banach spaces enjoy the Mazur-Ulam property. Following this line, in this manuscript it is proved that every reflexive or separable Banach space with dimension greater than or equal to 2 can be equivalently renormed to fail property P.
PB MDPI
SN 2227-7390
YR 2021
FD 2021-09
LK http://hdl.handle.net/10498/25832
UL http://hdl.handle.net/10498/25832
LA eng
NO This research was funded by Ministry of Science, Innovation and Universities of Spain, grant number PGC-101514-B-I00; and by the 2014-2020 ERDF Operational Programme and by the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia, grant number FEDER-UCA18-105867.
DS Repositorio Institucional de la Universidad de Cádiz
RD 13-nov-2026