Calculate the maximum points where the distance between every two points is within a certain range

Question

The predefined boundary is defined as $ (len, width) $.

(i)How to find the maximum points where the distance between every two points, $(x_{1},y_{1})$ and $(x_{2},y_{2})$, is within a certain range $(min,max)$

$min < \sqrt{(x_{2}-x_{1})^2 + (y_{2}-y_{1})^2} < max$

(ii)How about if those points are located on the edge of rectangle, and also:

$r < \sqrt{(x_{2}-x_{1})^2 + (y_{2}-y_{1})^2} < 2r$

Answer 1

If $min = 0$, the sphere is probably the shape with maximum volume in any number of dimensions. But in two dimensions at least, there are other maximal shapes $-$ shapes that don't admit the addition of any more points. These are the curves of constant width, or Reuleaux polygons $-$ see this Wikipedia article for definitions and examples.

If $min = max$, in $n$ dimensions the unique maximal solution is the vertices of a simplex, with $n+1$ points.

If $0 < min < max$, then the number of points must be finite; that's all I can think of for now.

Answer 2

If $\min=0$ it sounds like a sphere.

If $\min >0$ it sounds like some kind of Sierpinski gasket.