×

Disposition of points on a sphere of a Banach space and modulus of continuity of a function. (English. Russian original) Zbl 0598.46012

Math. Notes 35, 25-32 (1984); translation from Mat. Zametki 35, No. 1, 43-53 (1984).
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.

MSC:

46B20 Geometry and structure of normed linear spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Clarkson, ?Uniformly convex spaces,? Trans. Am. Math. Soc.,40, 396-414 (1936). · Zbl 0015.35604
[2] O. Hanner, ?On the uniform convexity of Lp andl p,? Arch. Math.,3, 239-244 (1956). · Zbl 0071.32801
[3] M. Day, ?Uniform convexity in factor and conjugate spaces,? Ann. Math.,45, 375-385 (1944). · Zbl 0063.01058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.