For a 2D closed convex shape, with metric $d$ and fixed area $A$, we can calculate the average distance between random (interior) points. For different shapes, we will get different values for this average. We can ask, for which shape is this average a minimum (in terms of $A$). For the Euclidean metric, the answer is the circle. If we pick a different metric, what shape do we get?

Here is an answer for lattice points and using the $L_1$ (Manhattan) metric:

Demaine, Erik D., Sándor P. Fekete, Günter Rote, Nils Schweer, Daria Schymura, and Mariano Zelke. "Integer point sets minimizing average pairwise $L_1$ distance: What is the optimal shape of a town?."

Computational Geometry44, no. 2 (2011): 82-94. PDF download.

The continuous version of the same problem is addressed in this paper:

Bender, Carl M., Michael A. Bender, Erik D. Demaine, and Sándor P. Fekete. "What is the optimal shape of a city?."

Journal of Physics A: Mathematical and General37, no. 1 (2003): 147. Journal link.

From the Abstract: "In this paper a nonlinear differential equation for the boundary curve of such a city is determined."

For an arbitrary metric and measure, this is going to be a very difficult question to answer. The shape of the minimizer will depend on the choice of metric and measure, if it exists at all. Joseph O'Rourke discussed the $L^1$ distance already, so I'm going to focus on distance functions which are either symmetric, or else induced by a smooth Riemannian metric. These sorts of questions are known as extremal problems [1] and are closely related to isoperimetric inequalities, which are an active area of research.

As noted in the comments, we need to specify which measure we use to compute averages. It appears that there are at least two natural choices of measure.

- the Lebesgue measure on $\mathbb{R}^2$
- the measure induced by a Riemannian metric

As I mentioned earlier, the particular shape of the minimizer will depend on the choice of distance function and measure. However, from a variational standpoint, there is a fairly general principle which is very helpful.

A convex region $S$ is a critical point for the average distance between interior points iff the integral $\int_S d(p,x) \, d \mu(x)$ is constant for all $p \in \partial S$. Here, $\mu$ is the measure used to determine averages.

In other words, a region $S$ is a possible minimizer for the average distance between points if the average distance between a point $p$ on the boundary of $S$ and interior points in $S$ does not depend on $p$. It's possible to prove this rigorously using variational techniques or using Crofton's differential equation (for a good exposition, see [2]). However, the basic intuition is that if this property does not hold, we can slightly change the shape of $S$ so as to reduce the total average distance.

From this, there are a few immediate consequences.

If the distance function is translation and rotationally symmetric and the measure we use is the Lebesgue measure on $\mathbb{R}^2$, the disk is a critical point. Oftentimes, this will be the unique critical point (and thus also the minimum), but any proof of this seems likely to depend on the particular details of the distance function.

If we consider a surface of constant curvature with its natural distance function and induced measure, the critical points for the average distance between points are geodesic balls. There is no guarantee that geodesic balls will be convex in the choice of coordinates you use, but they will be the minimizers nonetheless.

For more general (non-symmetric) Riemannian metrics (with their induced distance function and measure), this question is much more difficult, and the variational principles are less useful. However, with a lower bound on the sectional curvature, it is possible to get a lower bound on the areas of geodesic balls. Using this, it should be possible to establish uniform lower bound on the average distance between points in a region of area $A$. However, these sorts of estimates don't tell you what the minimizer looks like (or even if it exists at all). It is possible to construct distance functions where no minimizer exists, by incorporating patches where the curvature is more and more negative.

[1] *Bauer, Christina; Schneider, Rolf*, **Extremal problems for geometric probabilities involving convex bodies**, Adv. Appl. Probab. 27, No. 1, 20-34 (1995). ZBL0827.52004.

[2] *Eisenberg, Bennett; Sullivan, Rosemary*, **Crofton’s differential equation**, Am. Math. Mon. 107, No. 2, 129-139 (2000). ZBL0986.60011.

isthe metric. $\endgroup$ – Michael Mc Gettrick Feb 27 at 15:00