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.