The longest shortest fence and the stability of floating trees
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Free Boundary Problems and Related Topics
Over 50 years ago Polya stated the following problem. Given a plane convex set K (a piece of land), nd the shortest curve (or fence) that bisects this set into two subsets of equal area. Is it true that this curve is never longer than the diameter of the circular disc of same area as K? Under the additional assumption that K is centrosymmetric (i.e, K = K) he gave a simple proof that this is indeed the case. Without this assumption the question is much harder to answer positively. This is joint work with L. Esposito, V.Ferone, C. Nitsch and C. Trombetti. By the way, a result of N. Fusco and A. Pratelli states, that if the
fences are restricted to be straight line segments, the answer is negative. In that case the longest shortest fence is attained for the Auerbach triangle and not for the disc.
This talk is part of the Isaac Newton Institute Seminar Series series.
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