Exposing functions of strongly exposed points: Description and properties (Q1335919)

A point \(x\) in the unit sphere of a Banach space \(X\) is called strongly exposed if there is a continuous linear functional \(f\) on \(X\) with \(\| f\|= \langle f, x\rangle= 1\), and a function \(\phi: [0, 2]\to \mathbb{R}^ 1\) so that \(\phi(t)> 0\) if \(t> 0\) and \(f(y)\leq 1- \phi(\| y- x\|)\) for all \(\| y\|= 1\). In this case the function \(\phi\) is called an exposing function. The problem considered here is that of describing all exposing functions. The main result is the following partial solution: Let \(W\) denote the set of all functions \(w: [0,{1\over 2}]\to \mathbb{R}^ 1\) satisfying the conditions: 1) \(w(0)= 0\) and \(w(t)> 0\) if \(t> 0\), 2) \(w(kt)\geq kw(t)\) if \(k> 1\), 3) \(w(t)\leq t\). Then if \(F\) denotes the set of all exposing functions (for all possible Banach spaces), there are positive constants \(c_ 1\) and \(c_ 2\) for which \(c_ 1< c_ 2\) and \(c_ 1 W\subset F\subset c_ 2 W\). In addition, the author derives a number of interesting properties of exposing functions and considers the case of exposing functions for the special case of projective tensor products of Banach spaces.