Optimal Placement of Points inside a Set Say I have to place N points in $\mathbf{R}^2$ inside a circle of radius $R$.
I want to position them so as to maximize the sum of nearest distances i.e. solve the following problem
\begin{align}
\max_{x_1, ... x_n} &\sum_{i = 1}^{n} \min_{j \neq i} |x_i - x_j|^s \\
\text{s.t. } |x_i| &< R 
\end{align} 
 where $s$ is a constant larger than $2$ 
My aim is to characterize the behavior of this optimal value as the number of points $n$ becomes large. So even asymptotic order analysis will be helpful. 
If it makes things easier, consider other constraint sets (eg. A square of side length $A$)
 A: Not an exact solution, but an approximation (that may or not not be good enough):
Assumption 1: Solving the problem for the infinite plane provides a solution that is a reasonable approximation for the circle problem.
Assumption 2: The solution is somewhat uniform; for all points $x$, the distance to its nearest neighbor is $\delta$.
This corresponds to the problem of packing $n$ circles with radius $\frac{1}{2}\delta$ and center $x_i$. The densest solution is placing the $x_i$ on the vertices of a triangular tiling with equilateral triangles.
One way to construct this tiling inside a circle is to chose $\delta$ such that $\delta$ divides $r$ and to place the first vertex $x_0$ in the center of the circle. Add two more vertices to create the first triangle (with arbitrary orientation), then add the remaining triangles and discard all vertices that are not on/in the circle. This creates a neat, symmetric result, but has the disadvantage that $n$ results from $r$ and $\delta$, and cannot be given.
This solution is not optimal (look at the vertices closest to the circle perimeter), but for large $n$ it should approach the optimum.
A: You can use Lloyd's relaxation:


*

*Place the points randomly inside the circle.

*Iterate until convergence:
a) Compute the clipped Voronoi diagram of the points and the circle
b) Move each point to the center of mass of its Voronoi cell


Comments:
In 2a), the clipped Voronoi diagram is the intersection of the circle and the "ordinary" Voronoi diagram of the points. However, the available methods that compute it, usually assume $L_2$ norm (you mentioned $s > 2$, so be careful).
In 2b), compute the center of mass with respect to the distance you are using (whatever $s$ you have).
This approach has the advantage that you only specify the number of points $n$. In the related circle packing and Poisson disk sampling problems, you provide the circle radius or the desired minimal inter-point distance.
Reference:
Here is a link to a paper for computing the clipped Voronoi diagram (there are planty more on Google).
