While it is true that you don’t need a metric to draw a boundary, I personally need a metric to be able to envision high concentrations of probability density.
A concentration implies a region, which implies a metric space. While your sphering of the space normalises it somewhat and deals with part of the trouble, it still skips over the question of metric space. For example is 2, 2, 2 closer to 1, 1, 1 than 4, 1, 1? If that was a co-ordinate of a position in three dimensional space you would want to use the euclidean metric i.e. d = ((x2 - x1)^2 + (y2 - y1)^2+ (z2 - z1)^2)^1/2 or you that might not be appropriate and you would have to use city block distances and put them equally far away (if they were average energy usage, weight and how many copies of the gene for green eyes it had).
While it is true that you don’t need a metric to draw a boundary, I personally need a metric to be able to envision high concentrations of probability density.
A concentration implies a region, which implies a metric space. While your sphering of the space normalises it somewhat and deals with part of the trouble, it still skips over the question of metric space. For example is 2, 2, 2 closer to 1, 1, 1 than 4, 1, 1? If that was a co-ordinate of a position in three dimensional space you would want to use the euclidean metric i.e. d = ((x2 - x1)^2 + (y2 - y1)^2+ (z2 - z1)^2)^1/2 or you that might not be appropriate and you would have to use city block distances and put them equally far away (if they were average energy usage, weight and how many copies of the gene for green eyes it had).
See this page for more possible metrics http://www.cut-the-knot.org/do_you_know/far_near.shtml.