# The Longest Shortest Fence and Sharp Poincare-Sobolev Inequalities

created by kawohl on 29 Nov 2010

[BibTeX]

Preprint

Inserted: 29 nov 2010

Year: 2010

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.