The longest shortest fence and sharp Poincaré-Sobolev inequalities
type: Item
instance of: scholarly article
title: The longest shortest fence and sharp Poincaré-Sobolev inequalities
zbMATH Open document ID: 1262.52001
published in: Archive for Rational Mechanics and Analysis
publication date: 2012-11-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.6248
review text: The goal of this paper is to obtain sharp inequalities involving quantities which are related to the best way of halving planar convex sets. Problems of this type are called fencing problems. The first main result of this paper is the following theorem. If \(K\) is an open convex set of \(\mathbb{R}^2\), then \[ \inf_{G \subset K,\,|G|=|K|/2} \mathrm{Per}(G, K)^2 \leq \frac{4}{\pi}|K|, \] where \(|K|\) is the area of \(K\) and \(\mathrm{Per}(G, K)\) is the relative perimeter of \(G\) with respect to \(K\). The equality sign holds if and only if \(K\) is a disc. The second main result is the following theorem. Only the Auerbach triangle minimizes area in the class of convex sets whose shortest area-bisecting chord has given length.
reviewed by: Monika Nowicka
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A38
zbMATH DE Number: 6110421
zbMATH Keywords: fencing problems
zbMATH Keywords: planar convex sets
zbMATH Keywords: relative perimeter
author: Luca Esposito
author: Vincenzo Ferone
author: Bernd Kawohl
author: Carlo Nitsch
author: Cristina Trombetti
MaRDI profile type: Publication
OpenAlex ID: W1973571031
arXiv ID: 1011.6248
cites work: A remark on comparison results via symmetrization
cites work: Sur un problème de M. Ulam concernant l'équilibre des corps flottants
cites work: Q5506165
cites work: A relative isoperimetric inequality
cites work: Q3828608
cites work: A sharp form of Poincaré type inequalities on balls and spheres
cites work: Unsolved problems in geometry
cites work: On the geometric dilation of closed curves, graphs, and point sets
cites work: The Maximal Length of Chords Bisecting the Area Or Perimeter Length of Plane Convex Sets
cites work: Normal and integral currents
cites work: On a conjecture by Auerbach
cites work: An application of eigenfunctions of \(p\)-Laplacians to domain separation
cites work: Domain separation by means of sign changing eigenfunctions ofp-laplacians
cites work: Q3593480
cites work: Chords halving the area of a planar convex set
cites work: Rearrangements and convexity of level sets in PDE
cites work: The Longest Shortest Piercing
cites work: An inequality between the parts into which a convex body is divided by a plane section
cites work: Q5287551
cites work: Q4953817
cites work: SHARP POINCARÉ–SOBOLEV INEQUALITIES AND THE SHORTEST LENGTH OF SIMPLE CLOSED GEODESICS ON A TOPOLOGICAL TWO SPHERE
DOI: 10.1007/S00205-012-0545-0
description: scientific article