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Mappings without fixed or antipodal points. Some geometric applications. (English) Zbl 0967.46014

For \(T\) a topological space and \(X\) a real normed space \(S(T,X)\) denotes the set of continuous mappings from \(T\) into \(S(X)= \{x\in X:\|x\|= 1\}\). By \(C(T,X)= Y\) we denote the space of continuous and bounded mappings from \(T\) into \(X\) with usual uniform norm. If for every \(f\in S(T,X)\) exists a function \(e\in S(T,X)\) such that \(f(t)\neq e(t)\neq -f(t)\) for all \(t\in T\), then we say that \(S(T,X)\) is plentiful.
In the present paper the authors establish sufficient conditions for \(S(T,X)\) to be plentiful and study the extremal structure of the closed unit ball of \(Y\) in case \(X\) is strictly convex and \(S(T,X)\) is plentiful. The authors give an optimal representation of the points in \(B(Y)\) [= the closed unit ball of \(Y\)] as convex combination (and mean) of three extreme points when \(S(T,X)\) is plentiful.
For \(T\) completely regular and \(\dim X<+\infty\) they prove the following: the necessary and sufficient condition for every \(f\) in \(B(Y)\) to be the mean of three extreme points is that \(\dim T<\dim X\). If \(X\) is infinite-dimensional, then the previously mentioned representation remains true without any restriction about \(T\).

MSC:

46B20 Geometry and structure of normed linear spaces
46E15 Banach spaces of continuous, differentiable or analytic functions
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