Convexity and unique minimum points
type: Item
instance of: scholarly article
title: Convexity and unique minimum points
zbMATH Open document ID: 1441.03045
author: Josef Berger
author: Gregor Svindland
published in: Archive for Mathematical Logic
publication date: 2019-01-24T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03F60
Mathematics Subject Classification ID: 52A41
zbMATH DE Number: 7006124
zbMATH Keywords: Bishop's constructive mathematics
zbMATH Keywords: convex sets and functions
zbMATH Keywords: supporting hyperplanes
zbMATH Keywords: approximation theory
reviewed by: Daniel Abraham Romano
MaRDI profile type: Publication
full work available at URL: s00153-018-0619-2
OpenAlex ID: W2788929634
cites work: Brouwer's fan theorem and unique existence in constructive analysis
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cites work: Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallée Poussin's proof for Chebycheff approximation
DOI: 10.1007/S00153-018-0619-2
description: scientific article; zbMATH DE number 7006124