The point which is median in x and in y is the one that is natural
in such distance measurees as L1 which are not rotation-invariant.
L2 distance is the only Lp distance which is rotation invariant.
A notion of median friendly wth L2 distance is this:
a point is a "2D median" if and only if,
for every line thru that point which misses the others,
exactly half the other points are
on each side of it.
Note: such a median point may not exist.
BUT, if it does exist, then it has some nice properties.
In particular, it is a Condorcet winner for voters (thepoints) who
rank the candidates in decreasing order of L2 distance away from them.
So this is what I meant when I said Condorcet voting was actually friendlier
with L2 than with L1 distance, contrary to unfounded myths.
By the way, when Scott Ritchie redrew my counterexample with rotated axes, he
was
right that the "median" with respect to the two new axial directions moves to
X. However, you can set up the counterexample so that it does not.
(You have to move the voters around a bit.) I do not know how nasty
you can make this counterexample get in terms of making it immune to
different angles of axis rotation. Can we make it immune to all of them?
Warren D Smith
http://rangevoting.org
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