McMullen problem

The McMullen problem is an open problem in discrete geometry named after Peter McMullen.

In 1972, David G. Larman wrote about the following problem:[1] Larman credited the problem to a private communication by Peter McMullen.

Using the Gale transform, this problem can be reformulated as: The numbers

of the original formulation of the McMullen problem and

of the Gale transform formulation are connected by the relationships

Also, by simple geometric observation, it can be reformulated as: The relation between

The equivalent projective dual statement to the McMullen problem is to determine the largest number

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.

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.