The longest shortest fence and sharp Poincaré-Sobolev inequalities
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.
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