Convexity and unique minimum points (Q1712926)

In this paper, as a continuation of previously their published paper [Arch. Math. Logic 55, No. 7--8, 873--881 (2016; Zbl 1402.03094)] the authors proved in a very inventive way that every quasi-convex, uniformly continuous function \(f : C \longrightarrow \mathbb{R}\) with at most one minimum point has a minimum point, where C is a convex compact subset of a \(\mathbb{R}^{n}\). Extending the previous theorem to finite-dimensional normed space, as applications of such generalized theorem they obtain a supporting hyperplane theorem, a result on strictly quasi-convex functions and a new short proof of the fundamental theorem of approximation theory [\textit{D. S. Bridges}, Rocky Mt. J. Math. 11, 491--497 (1981; Zbl 0478.41021)]. The article is written concisely, elegantly and very legibly.