Disposition of points on a sphere of a Banach space and modulus of continuity of a function (Q1079780)

Let X be a uniformly Banach space, \(S_ R=\{x\in X:\) \(\| x\| =R\}\) be the sphere of radius R with center at 0, and \(\delta (u)=\inf_{\| x\| =\| y\| =1, \| x-y\| =u}(1-\| (x+y)/2\|)\) be the modulus of convexity of X. Let us consider the following problem: There are given three points x, y, and z on \(S_ R\) such that \(\| x-y\| \leq \epsilon\) and \(\| y- z\| \leq \epsilon\), where \(\epsilon >0\). How much near x and z are ? Since x, y, and z belong to a sphere of a uniformly convex space, the classical triangle inequality \[ \| x-z\| =\| x-y+y-z\| \leq \| x-y\| +\| y-z\| =2\epsilon \] gives a rough estimate. The exact answer \(\phi_ X(\epsilon,R)\) depends on \(\epsilon\) and R. When \(\epsilon\) is ''large'', \(\| x-z\| \leq 2R\) and when \(\epsilon\) is ''small'', \(\| x-z\| \leq \phi_ X(\epsilon,R)\), \(\phi_ X(\epsilon,R)\leq \sup_{u}\{u:\) \(R\delta (u/R)+\epsilon \delta (u/\epsilon)\leq \epsilon \}\). The critical value of \(\epsilon\) \((\epsilon_{cr})\), at which the change of estimates takes place, is not less than the root of the equation \(R/\epsilon +\delta (2R/\epsilon)=1\). Relying on Clarkson's inequalities, we compute and investigate \(\phi_ X(\epsilon,R)\) for \(X=L_ p\), \(1<p<\infty\). The obtained results give necessary conditions on the modulus of continuity of a function from \(L_ p\) and a partial answer to a problem of S. B. Stechkin.