[revised] Henry Baker <hbaker1@pipeline.com> sez:
Consider points more-or-less equally spaced around a circle in 2D. What should be the "median" ?
Well, somewhere close to the center anyway...? With the directions-sum-to-zero median, if you take, say, n > 2 points at equally-spaced angles but random distances from some point, you get that point as the median. E.g. the intersection of a cross.
If I cut the set with hyperplanes, I'd like the hyperplanes to cut the set in half. Unfortunately, the intersection of [sw: all of] these hyperplanes is likely to be empty.
Yup, can't be that. But maybe: project the points onto every line through the origin, and take the median for each of those lines, then repeat on the result until it converges. Of course this runs into that same counting problem I mentioned before. But I think it avoids the discontinuity problem. (Funny, it also resembles one way to think about a convex hull.) I think my sum of normalized vectors definition is still "median-like". It's the point where there's a "balance of directions." --Steve