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Continuity of the uniform rotundity modulus relative to linear subspaces. (English) Zbl 0886.46013
Let \((X,|\cdot |)\) be a Banach space. A modulus of rotundity relative to a linear subspace \(Y\subset X\) is defined as \(\delta(Y,\epsilon) = \inf \big \{1-{1\over 2}|x+y|:\;x,y\in X,\^^M|x|\leq 1,\;|y|\leq 1,\;x-y \in Y,\;|x-y|\geq \epsilon \big \},\;\^^M0\leq \epsilon \leq 2\). Let \(\mathcal S\) denote the family of the unit spheres \(S_Y\), where \(Y\) runs through all linear subspaces of \(Y\subset X\). Endow \(\mathcal S\) by the Hausdorff semimetric. Theorem. The function \(\delta : {\mathcal S}\times [0,2) \to [0,+\infty)\) is continuous. On two examples, it is shown that \(\delta(Y,\cdot)\) may not be continuous at \(2\) and that \(\delta(\cdot,2)\) may not be continuous.
Reviewer: M.Fabian (Praha)
46B20 Geometry and structure of normed linear spaces
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