Lipschitz maps and the geometry of the unit ball in normed spaces (Q1058108)

A new geometric property of normed spaces is introduced and investigated. For any point x of the unit sphere S of a normed space X and any v in X set: \(\tau_ x(v)=\sup \{f(v),f\in J(x)\}\) where \(J(x)=\{f\in S^*: f(x)=1\};\) for \(k\geq 1\), \(x\in S\) and \(t\geq 0\) define \[ \beta_{x,k}(t)=\inf \{(1-\| x+tv\|:\tau_ x(v)\leq -1,\| v\| \leq k\}; \] the following is the main definition: x is said to be a \(\tau\)-point if for any \(k\geq 1\) there is a \(\sigma =\sigma (x,k)>0\) such that \(\beta_{x,k}(\sigma)>0.\) The global version (\(\theta\)-property) and the uniform global version (T- property) of the \(\tau\)-property are also defined and studied. In particular it is shown that every finite dimensional and every uniformly smooth space has property T. A complete characterization of \(\tau\)-point is given in some classical Banach space. The space \(c_ 0\) of null sequences is shown to have the \(\theta\)-property and to fail the T- property. Applications are given to the study of Lipschitz self-maps of the unit ball of X. In particular, it is proved that if X has propety T, then holds uniformly an upper bound for the minimal displacement of k- Lipschitz self maps on the unit ball sharper than 1-1/k. This gives new information on a question raised by K. Goebel.