Optimal shape of a blob

This paper presents the solution to the following optimization problem: What is the shape of the two-dimensional region that minimizes the average $Lp$ distance between all pairs of points if the area of this region is held fixed? Variational techniques are used to show that the boundary curve of the optimal region satisfies a nonlinear integral equation. The special case $p=2$ is elementary and for this case the integral equation reduces to a differential equation whose solution is a circle. Two nontrivial special cases, $p=1$ and $p=∞$⁠, have already been examined in the literature. For these two cases the integral equation reduces to nonlinear second-order differential equations, one of which contains a quadratic nonlinearity and the other a cubic nonlinearity.