# zbMATH — the first resource for mathematics

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)
##### MSC:
 46B20 Geometry and structure of normed linear spaces
Full Text: