A further property of spherical isometries (Q2922944)

This paper consists of three sections. The introduction starts with a brief history of Tingley's property and its relation with the problem solved in the paper.NEWLINENEWLINEIn Section 2, the author defines the notion of the frame of the unit ball \(B_X\) for a real Banach space \(X\) and the notion of \(k\)-extreme points of \(B_X\) as follows. An exposed face of \(B_X\) is defined to be \(f^{-1}(\{1\})\cap B_X\) with \(f\) a supporting functional of \(B_X\), \(E(f)\) denotes the relative boundary of the exposed face with respect to the hyperplane \(f^{-1}(\{1\})\). The frame of \(B_X\) is defined to be \(\mathrm{frm}(B_X)=\{ E(f): f \, \text{is a supporting functional for } B_X\}\). A point \(x \in X\) is said to be a \(k\)-extreme point of \(B_X\) if and only if every \(k+1\) unit vectors with average equal to \(x\) are linearly dependent, and the set of \(k\)-extreme points is denoted by \(\mathrm{ext}_{k}(B_X)\). This section ends with three results relating these two concepts.NEWLINENEWLINEIn the last section, the author proves two main theorems. The first one gives a simple characterization of \(\mathrm{frm}B_X\). If \(x\) is a unit vector, then \(x \notin \mathrm{frm}(B_X)\) if and only if \((x+tB_X)\cap S_X\) is convex for some positive number \(t\). The second theorem reads as follows:NEWLINENEWLINE{Theorem.} Let \(X\) and \(Y\) be real Banach spaces. Suppose \(T_0:S_X \rightarrow S_Y\) is a surjective isometry. Then \(T_0(\mathrm{frm}(B_X))=\mathrm{frm}(B_Y)\). If the spaces are of dimension \(n\) and \(T_0:S_X \rightarrow S_Y\) is a surjective isometry, then \(T_0(\mathrm {ext}_{n-1}(B_X))=\mathrm{ext}_{n-1}(B_Y)\).NEWLINENEWLINEThe paper ends with some considerations on extensions of a surjective isometry defined between the frames of the unit balls of the spaces involved.