Geometric inequalities in normed spaces
type: Item
instance of: scholarly article
title: Geometric inequalities in normed spaces
zbMATH Open document ID: 0788.46017
published in: Rocky Mountain Journal of Mathematics
publication date: 1993-01-17T00:00:00Z
review text: Let \(x\) be a \(T\)-periodic solution of \(x'= f(x)\), where \(f: E\to E\) is Lipschitz continuous with constant \(L\) and \(E\) is a Banach space. In [the first author, \textit{D. Fisher} and the second author, Proc. Am. Math. Soc. 98, 376-378 (1986; Zbl 0632.34048)] it is shown that \(TL\geq 6\); this inequality is optimal [cf. the first author, \textit{D. Fisher} and the second author, Am. Math. Mon. 96, No. 1, 5-17 (1989; Zbl 0672.34034)]. The authors of the paper under review apply ideas and methods of the above mentioned works to solve some problems of geometrical nature. They obtain new proofs and/or new results which generalize the existing ones. Among other things, they extend the classical result of \textit{W. Fenchel} [Bull. Am. Math. Soc. 57, 44-54 (1951; Zbl 0042.400)] to any finite or infinite dimensional Hilbert space, they prove inequalities on sums of distances between finitely many vectors in normed spaces [special cases yield results from \textit{G. D. Chakerian} and \textit{M. S. Klamkin}, Am. Math. Mon. 80, 1009-1017 (1973; Zbl 0276.52005)], they improve an inequality concerning convex sets which was established by \textit{A. D. Andrew} and \textit{M. A. Ghandehari} [Congr. Numerantium 50, 31-35 (1985; Zbl 0592.52006)].
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46C05
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 97419
zbMATH Keywords: geometric inequalities
zbMATH Keywords: distance
zbMATH Keywords: convex hull
zbMATH Keywords: periodic solution
zbMATH Keywords: curvature of curves
zbMATH Keywords: Hamiltonian cycles of a graph
zbMATH Keywords: \(T\)-periodic solution
author: Stavros N. Busenberg
author: Mario Martelli
reviewed by: Bohumír Opic
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072742
OpenAlex ID: W2026832775
cites work: Q3721326
cites work: Q5795778
cites work: Bounds for the period of periodic orbits of dynamical systems
cites work: Shorter Notes: Better Bounds for Periodic Solutions of Differential Equations in Banach Spaces
cites work: Minimal Periods of Discrete and Smooth Orbits
cites work: Q5749618
cites work: Inequalities for Sums of Distances
cites work: Discrete analogs of inequalities of Wirtinger
cites work: On the differential geometry of closed space curves
cites work: On Fenchel's Theorem
cites work: Bounds for periodic solutions of differential equations in Banach spaces
cites work: Q4757601
cites work: Q3210352
cites work: Q5338702
cites work: Periods of Periodic Solutions and the Lipschitz Constant
DOI: 10.1216/RMJM/1181072742
description: scientific article; zbMATH DE number 97419