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Linear combinations of partitions of unity with restricted supports. (English) Zbl 1013.54008

It is shown that a \(T_1\)-space \(X\) is normal if and only if it has the property that for every locally finite open covering \({\mathcal U}= (U_i)_{i\in I}\) of \(X\), for every continuous mapping \(f\) from \(X\) into a Hausdorff topological vector space \(E\), and for every family \((y_i)_{i\in I}\) of vectors in \(E\) such that \(f(x)\) belongs to the convex hull of \(\{y_i\mid x\in U_i\}\) for each \(x\in X\) there exists a partition of unity \(\{\varphi_i\mid i\in I\}\) subordinate to \({\mathcal U}\) such that \(f= \sum_{i\in I}\varphi_i y_i\). This interesting characterization of normality is then used in optimization theory to determine the class of all functions \(f\in C(|{\mathcal P}|)\) on the underlying space \(|{\mathcal P}|\) of a Euclidean complex \({\mathcal P}\) such that, for each polytope \(P\in{\mathcal P}\), the restriction \(f|_P\) attains its extrema at vertices of \(P\). Moreover, a class of extremal functions on the metric space \(([-1,1]^m, d_\infty)\) is characterized, which appears in approximation theory under the name controllable partitions of unity.

MSC:

54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
49J99 Existence theories in calculus of variations and optimal control
54C65 Selections in general topology
52B11 \(n\)-dimensional polytopes
41A30 Approximation by other special function classes
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