The longest shortest fence and sharp Poincar\'e-Sobolev inequalities
Optimization and Control
2010-11-30 v1 Analysis of PDEs
Abstract
We prove a long standing conjecture concerning the fencing problem in the plane: among planar convex sets of given area, prove that the disc, and only the disc maximizes the length of the shortest area-bisecting curve. Although it may look intuitive, the result is by no means trivial since we also prove that among planar convex sets of given area the set which maximizes the length of the shortest bisecting chords is the so-called Auerbach triangle.
Keywords
Cite
@article{arxiv.1011.6248,
title = {The longest shortest fence and sharp Poincar\'e-Sobolev inequalities},
author = {L. Esposito and V. Ferone and B. Kawohl and C. Nitsch and C. Trombetti},
journal= {arXiv preprint arXiv:1011.6248},
year = {2010}
}
Comments
30 pages, 12 figures