Geometric Invariants of Surjective Isometries between Unit Spheres

Almudena Campos-Jiménez,Francisco Javier García-Pacheco

doi:10.3390/math9182346

Abstract

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.

Highlights

The isometric extension problem is a prolific topic in the area of Functional Analysis inspired by the so-called Mazur-Ulam Theorem [1], which affirms that an isometry defined between two real Banach spaces is affine, understanding an isometry as a mapping T : X → Y preserving distances, that meansT(x) − T(y) = x − y for all x, y ∈ X
A notable convex body in Banach spaces is the unit ball, which motivates the step in the isometric extension problem: reducing the domain and co-domain to the unit spheres of both spaces
Is it always possible to extend a surjective isometry defined between the unit spheres of two real Banach spaces to a surjective linear isometry between the whole spaces?

Summary

Introduction

A subtle generalization of this result is due to Mankiewicz [2], which states that it is sufficient to define the isometry between two convex bodies (convex sets with non-empty interior) of the real normed spaces to uniquely extend to an affine isometry between the whole spaces. A notable convex body in Banach spaces is the unit ball, which motivates the step in the isometric extension problem: reducing the domain and co-domain to the unit spheres of both spaces. This question is known as Tingley’s Problem, due to Tingley [3]. Is it always possible to extend a surjective isometry defined between the unit spheres of two real Banach spaces to a surjective linear isometry between the whole spaces?

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Conclusion