A two-dimensional inequality and uniformly continuous retractions.(English)Zbl 1149.46014

The authors consider the problem of expressing the identity mapping of the unit ball $$B_X$$ of an infinite-dimensional Banach space $$X$$ as a convex combination of uniformly continuous retractions from $$B_X$$ onto the unit sphere $$S_X$$. Problems of this type are related to the study of the extremal structure of the space of continuous functions taking values in the Banach space $$X$$. The authors prove that, if $$X$$ is an infinite-dimensional uniformly rotund Banach space, $$0<\alpha<1$$, $$(1-\alpha)/2 \leq \lambda \leq 1/2$$, and $$B$$ is the closed annulus with radii $$\alpha$$ and $$1$$, then the identity mapping on $$B$$ can be expressed as a convex combination $$\lambda r_1 + (1 - \lambda)r_2$$, where $$r_1, r_2:B\to S_X$$ are uniformly continuous retractions.
As a corollary, the authors obtain that if $$X$$ is an infinite-dimensional uniformly rotund Banach space and $$\lambda_1, \lambda_2, \lambda_3$$ lie strictly between $$0$$ and $$1/2$$ and sum to $$1$$, then there exist three uniformly continuous retractions $$r_1,r_2,r_3$$ from $$B_X$$ onto $$S_X$$ so that the identity map on $$B_X$$ can be expressed as the convex combination $$\lambda_1 r_1 + \lambda_2 r_2 + \lambda_3 r_3$$. The authors further note that a similar corollary holds for $$n$$ retractions for any $$n\geq 3$$ and that the hypothesis that the Banach space is uniformly rotund can be replaced by a more general property that is satisfied by any complex Banach space.

MSC:

 46B20 Geometry and structure of normed linear spaces 46E40 Spaces of vector- and operator-valued functions 54C15 Retraction
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References:

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