The almost surely shrinking yolk

The yolk, defined by McKelvey as the smallest ball intersecting all median hyperplanes, is a key concept in the Euclidean spatial model of voting. Koehler conjectured that the yolk radius of a random sample from a uniform distribution on a square tends to zero. The following sharper and more general results are proved here: Let the population be a random sample from a probability measure µ on <m. Then the yolk of the sample does not necessarily converge to the yolk of µ. However, if µ is strictly centered, i.e. the yolk radius of µ is zero, then the radius of the sample yolk will converge to zero almost surely, and the center of the sample yolk will converge almost surely to the center of the yolk of µ. Moreover, if the yolk radius of µ is nonzero, the sample yolk radius will not converge to zero if µ contains three noncollinear mass points or if somewhere it has density bounded away from zero in some ball of positive volume. All results hold for both odd and even population sizes.