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.


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