Given a locally finite open covering of a normal space X and a Hausdorff topological vector space E, we characterize all continuous functions f: X → E which admit a representation $f={\sum }_{C\in }{a}_C{\phi }_C$ with $a_C\in E$ and a partition of unity ${\phi }_C:C\in$ subordinate to . As an application, we determine the class of all functions f ∈ C(||) on the underlying space || of a Euclidean complex such that, for each polytope P ∈ , the restriction ${f|}_P$ attains its extrema at vertices of P. Finally, a class of extremal functions on the metric space...