Geometric Invariants of Surjective Isometries between Unit Spheres

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.

Compact convex sets free of inner points in infinite‐dimensional topological vector spaces

“…In [2, Definition 17], the following concepts were introduced in the literature of the Banach space geometry: let $$X$$ be a normed space. Let $$E\subseteq \mathsf {S}_X$$.…”

“…In [2, Lemma 6], the following essential properties were proved for a normed space $$X$$ and a subset $$E\subseteq \mathsf {S}_X$$: (1)If $$E$$ is convex, then $$E$$ is flat and starlike compatible. (2)$$E$$ is almost flat if and only if $$E$$ is starlike compatible. (3)If $$E$$ is flat, then $$E$$ is almost flat. (4)$$E$$ is flat if and only if $$\mathrm{co}(E)\subseteq \mathrm{st}(E)$$. (5)If $$E$$ is flat and $$D$$ is a convex component of $$\mathsf {S}_X$$ containing $$E$$, then $$D\subseteq \mathrm{st}(E)$$. (6)If $$E$$ is a convex component of $$\mathsf {S}_X$$, then $$E$$ is starlike generated. (7)If …”

“…A novel three‐dimensional unit ball is constructed in [2, Example 3] which serves to show the existence of an almost flat set which is not flat.…”

“…Proof First off, $$E$$ is clearly not convex. However, $$E$$ is trivially almost flat, thus $$E$$ is starlike compatible by [2, Lemma 6(2)], hence $$E\subseteq \mathrm{st}(E)$$. It only remains to show that $$\mathrm{st}(E)\subseteq E$$.…”

Exploring new solutions to Tingley's problem for function algebras

“…The problem of extending a surjective isometry between the unit spheres of two Banach spaces -named Tingley's problem after the contribution of D. Tingley in [45]is nowadays a trending topic in functional analysis (see a representative sample in the references [6,14,15,19,20,21,22,23,34,35,38,9] and the surveys [47,37]). This isometric extension problem remains open for Banach spaces of dimension bigger than or equal to 3 though.…”