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The longest shortest fence and sharp Poincar\'e-Sobolev inequalities

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arxiv 1011.6248 v1 pith:62FZFHZV submitted 2010-11-29 math.OC math.AP

L. Esposito , V. Ferone , B. Kawohl , C. Nitsch , C. Trombetti This is my paper · ORCID
classification math.OCmath.AP
keywords proveshortestareaconvexdiscgivenlengthmaximizes
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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.

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