Editing Median voter theorem (section)

===Relation between the median in all directions and the geometric median===
Whenever a unique omnidirectional median exists, it determines the result of Condorcet voting methods. At the same time the [[Median#Multivariate median|geometric median]] can arguably be identified as the ideal winner of a ranked preference election. It is therefore important to know the relationship between the two. In fact whenever a median in all directions exists (at least for the case of discrete distributions), it coincides with the geometric median.

[[File:Median Plott.svg|thumb|Diagram for the lemma]]'''''Lemma'''''. Whenever a discrete distribution has a median ''M''&nbsp; in all directions, the data points not located at ''M''&nbsp; must come in balanced pairs (''A'',''A''&nbsp;'&nbsp;) on either side of ''M''&nbsp; with the property that ''A''&nbsp;–&nbsp;''M''&nbsp;–&nbsp;''A''&nbsp;' is a straight line (ie. ''not'' like ''A''<sub>&nbsp;0&nbsp;</sub>–&nbsp;''M''&nbsp;–&nbsp;''A''<sub>&nbsp;2</sub> in the diagram).

'''''Proof'''''. This result was proved algebraically by Charles Plott in 1967.<ref>C. R. Plott, "A Notion of Equilibrium and its Possibility Under Majority Rule" (1967).</ref> Here we give a simple geometric proof by contradiction in two dimensions.

Suppose, on the contrary, that there is a set of points ''A<sub>i</sub>'' which have ''M''&nbsp; as median in all directions, but for which the points not coincident with ''M''&nbsp; do not come in balanced pairs. Then we may remove from this set any points at ''M'', and any balanced pairs about ''M'', without ''M''&nbsp; ceasing to be a median in any direction; so ''M''&nbsp; remains an omnidirectional median.

If the number of remaining points is odd, then we can easily draw a line through ''M''&nbsp; such that the majority of points lie on one side of it, contradicting the median property of ''M''.

If the number is even, say 2''n'', then we can label the points ''A''<sub>&nbsp;0</sub>, ''A''<sub>1</sub>,... in clockwise order about ''M''&nbsp; starting at any point (see the diagram). Let θ be the angle subtended by the arc from ''M''&nbsp;–''A''<sub>&nbsp;0</sub> to ''M''&nbsp;–''A''<sub>&nbsp;''n''&nbsp;</sub>. Then if θ&nbsp;&lt;&nbsp;180° as shown, we can draw a line similar to the broken red line through ''M''&nbsp; which has the majority of data points on one side of it, again contradicting the median property of ''M''&nbsp;; whereas if θ&nbsp;&gt;&nbsp;180° the same applies with the majority of points on the other side. And if θ&nbsp;=&nbsp;180°, then ''A''<sub>&nbsp;0</sub> and ''A''<sub>&nbsp;''n''</sub> form a balanced pair, contradicting another assumption.

'''''Theorem'''''. Whenever a discrete distribution has a median ''M''&nbsp; in all directions, it coincides with its geometric median.

'''''Proof'''''. The sum of distances from any point ''P''&nbsp; to a set of data points in balanced pairs (''A'',''A''&nbsp;'&nbsp;) is the sum of the lengths ''A''&nbsp;–&nbsp;''P''&nbsp;–&nbsp;''A''&nbsp;'. Each individual length of this form is minimized over ''P'' when the line is straight, as happens when ''P''&nbsp; coincides with ''M''. The sum of distances from ''P'' to any data points located at ''M'' is likewise minimized when ''P''&nbsp; and ''M''&nbsp; coincide. Thus the sum of distances from the data points to ''P'' is minimized when ''P'' coincides with ''M''.