Maximum area with Minkowski measures of perimeter

Abstract

The oldest competition for an optimal (area-maximizing) shape was won by the circle. But if the fixed perimeter is measured by the line integral of $|\mathrm{d} x|+|\mathrm{d} y|$, a square would win. Or if the boundary integral of $\max(|\mathrm{d} x|,|\mathrm{d} y|)$ is given, a diamond has maximum area. For any norm in $\mathbb{R}^2$, we show that when the integral of $\|(\mathrm{d} x,\mathrm{d} y)\|$ around the boundary is prescribed, the area inside is maximized by a ball in the dual norm (rotated by $\pi/2$).

This ‘isoperimetrix' was found by Busemann. For polyhedra it was described by Wulff in the theory of crystals. In our approach, the Euler–Lagrange equation for the support function of $S$ has a particularly nice form. This has application to computing minimum cuts and maximum flows in a plane domain.