On a question of Gromov about the Wirtinger inequalities
type: Item
instance of: scholarly article
title: On a question of Gromov about the Wirtinger inequalities
zbMATH Open document ID: 1398.53045
author: Takefumi Kondo
author: Tetsu Toyoda
author: Takato Uehara
published in: Geometriae Dedicata
publication date: 2018-08-23T00:00:00Z
review text: This paper studies various inequalities on metric spaces: Gromov's \(\mathrm{Cycl}_k(0)\) conditions and the Wirtinger inequalities \(\mathrm{Wir}_k\). Gromov showed that for any \(k \geq 4\), the property of being \(\mathrm{CAT}(0)\) implies \(\mathrm{Cycl}_k(0)\), which in turn implies \(\mathrm{Wir}_k\). On the other hand, for a geodesic space, the \(\mathrm{Cycl}_4(0)\) condition implies \(\mathrm{CAT}(0)\), and so for geodesic spaces the implication \(\mathrm{Cycl}_4(0) \implies \mathrm{Wir}_k\) holds for all \(k \geq 4\). Does this statement, \(\mathrm{Cycl}_4(0) \implies \mathrm{Wir}_k\), hold without the assumption that the space is geodesic? This paper answers this question of Gromov in the affirmative, using Perron-Frobenius theory.
reviewed by: John Harvey
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 51F99
zbMATH DE Number: 6924049
zbMATH Keywords: CAT(0) space
zbMATH Keywords: Wirtinger space
zbMATH Keywords: \(\mathrm {Cycl}_k(0)\) space
MaRDI profile type: Publication
full work available at URL: s10711-017-0284-3
OpenAlex ID: W2761110373
cites work: Q4848742
cites work: Quasilinearization and curvature of Aleksandrov spaces
cites work: Q4720067
cites work: On the nonexistence of uniform homeomorphisms between \(L^ p\)-spaces
cites work: Napoleon's Theorem and the Parallelogram Inequality for Affine-Regular Polygons
cites work: \(\text{CAT}(\kappa)\)-spaces: construction and concentration
cites work: Q4380377
cites work: Roundness properties of groups.
cites work: A note on Markov type constants
cites work: Q3582169
cites work: Inequality between sides and diagonals of a space $n$-gon and its integral analog
cites work: An alternative proof of Berg and Nikolaev's characterization of \(CAT(0)\)-spaces via quadrilateral inequality
cites work: The Finite Fourier Series and Elementary Geometry
cites work: Q3857416
cites work: Q4087773
cites work: Matrix theory
DOI: 10.1007/S10711-017-0284-3
description: scientific article; zbMATH DE number 6924049