An isoperimetric deficit formula
type: Item
instance of: scholarly article
title: An isoperimetric deficit formula
zbMATH Open document ID: 1470.53047
author: Peter McGrath
published in: The Journal of Geometric Analysis
publication date: 2021-05-04T00:00:00Z
review text: Let \(\Omega\) be a bounded domain with \(C^1\) boundary in either \(\mathbb{R}^2,\mathbb{S}^2\) or \(\mathbb{H}^2\). Let \(K\) be the Gaussian curvature of the ambient space, let \(L\) be the lenght of \(\partial \Omega\) and let \(A\) be the area of \(\Omega\). The author shows that the isoperimetric deficit \(L^2-4\pi A+KA^2\) can be expressed as an integral comparing the outward normal field along \(\partial \Omega\) at two points. The formula is shown in the Euclidean case and then extended to the non-zero curvature case.
reviewed by: Alessio Savini
Mathematics Subject Classification ID: 53C24
zbMATH DE Number: 7343462
zbMATH Keywords: isoperimetric inequality
zbMATH Keywords: Bonnesen
zbMATH Keywords: isoperimetric deficit
MaRDI profile type: Publication
full work available at URL: s12220-020-00393-3
OpenAlex ID: W3011153927
cites work: A generalization of the isoperimetric inequality
cites work: A quantitative isoperimetric inequality on the sphere
cites work: The Isoperimetric Theorem for Curves on Minimal Surfaces
cites work: The sharp quantitative isoperimetric inequality
cites work: A strong form of the quantitative isoperimetric inequality
cites work: Inégalité isopérimétrique et calibration. (Isoperimetric inequality and calibration)
cites work: Bonnesen-Style Isoperimetric Inequalities
cites work: Zwei kurze Beweise der isoperimetrischen Ungleichung
cites work: On the isoperimetric inequality on a minimal surface
DOI: 10.1007/S12220-020-00393-3
Recommended article: A note on the isoperimetric deficit
Recommended article: An integral form of the isoperimetric inequality
Recommended article: Q3743971
Recommended article: A lower bound for the isoperimetric deficit
Recommended article: An error estimate for the isoperimetric deficit
Recommended article: Inequalities for dual isoperimetric deficits
Recommended article: The isoperimetric method
Recommended article: Q5693288
description: scientific article; zbMATH DE number 7343462