OK, a nastier median+L1 counterexample is as follows. 2 Candidates: X = (8,8) O = (0,0)
8 Voters: (-1,-1) (-2, -2) (-3, -3) (4+e, 4) (4, 4+e) (19, 19) (-4-B, 12+e+B) (12+e+B, -4-B) where e>0 is a very small number and B>0 is a very big number. Optionally: place an additional 9th voter exactly at O. O is the median among the candidates in both x-coordinate and y-coordinate, but X beats O in a majority vote election where voters prefer candidates closer in L1 distance (or in Lp distance for any p>=1). If the axes are rotated by a random angle, then O is still the median in both axial directions, with probability 0.9999 (can be made arbitrarily near 1 but not 1 itself). Also, I think X always beats O in the election no matter what angle you rotate the coords. (Also: There is no "2D median" of this point set.) Warren D Smith http://rangevoting.org ---- election-methods mailing list - see http://electorama.com/em for list info
