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Extreme points and retractions in Banach spaces. (English) Zbl 0901.46015
Summary: For $$T$$ a completely regular topological space and $$X$$ a strictly convex Banach space, we study the extremal structure of the unit ball of the space $$C(T,X)$$ of continuous and bounded functions from $$T$$ into $$X$$. We show that when $$\dim X$$ is an even integer then every point in the unit ball of $$C(T,X)$$ can be expressed as the average of three extreme points if, and only if, $$\dim T<\dim X$$, where $$\dim T$$ is the covering dimension of $$T$$. We also prove that, if $$X$$ is infinite-dimensional, the aforementioned representation of the points in the unit ball of $$C(T,X)$$ is always possible without restrictions on the topological space $$T$$. Finally, we deduce from the above result that the identity mapping on the unit ball of an infinite-dimensional strictly convex Banach space admits a representation as the mean of three retractions of the unit ball onto the unit sphere.

##### MSC:
 46B20 Geometry and structure of normed linear spaces 46E40 Spaces of vector- and operator-valued functions
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##### References:
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