Larman credited the problem to a private communication by Peter McMullen.
Equivalent formulations
Gale transform
Using the Gale transform, this problem can be reformulated as:
Determine the smallest number such that for every set of points
in linearly general position on the sphere it is possible to choose a set where for , such that every open hemisphere of contains at least two members of .
The numbers of the original formulation of the McMullen problem and of the Gale transform formulation are connected by the relationships
Partition into nearly-disjoint hulls
Also, by simple geometric observation, it can be reformulated as:
Determine the smallest number such that for every set of points in there exists a partition of into two sets and with
The relation between and is
Projective duality
An arrangement of lines dual to the regular pentagon. Every five-line projective arrangement, like this one, has a cell touched by all five lines. However, adding the line at infinity produces a six-line arrangement with six pentagon faces and ten triangle faces; no face is touched by all of the lines. Therefore, the solution to the McMullen problem for d = 2 is ν = 5.
The equivalent projective dual statement to the McMullen problem is to determine the largest number such that every set of hyperplanes in general position in d-dimensional real projective space form an arrangement of hyperplanes in which one of the cells is bounded by all of the hyperplanes.
Results
This problem is still open. However, the bounds of are in the following results: