McMullen problem

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

Statement

In 1972, David G. Larman wrote about the following problem:Template:R Template:Block indent 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: Template:Block indent

The numbers $\nu$ of the original formulation of the McMullen problem and $\mu$ of the Gale transform formulation are connected by the relationships $$\begin{array}{cc}\mu (k)& =\min \{w\mid w\le \nu (w-k-1)\}\\ \nu (d)& =\max \{w\mid w\ge \mu (w-d-1)\}\end{array}$$

Partition into nearly-disjoint hulls

Also, by simple geometric observation, it can be reformulated as: Template:Block indent

The relation between $\mu$ and $\lambda$ is $$\mu (d+1)=\lambda (d),d\ge 1$$

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 $\nu (d)$ such that every set of $\nu (d)$ 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 $\nu (d)$ are in the following results:

• David Larman proved in 1972 thatTemplate:R $$2d+1\le \nu (d)\le (d+1{)}^2.$$
• Michel Las Vergnas proved in 1986 thatTemplate:R $$\nu (d)\le \frac{(d+1)(d+2)}{2}.$$
• Jorge Luis Ramírez Alfonsín proved in 2001 thatTemplate:R $$\nu (d)\le 2d+\lceil \frac{d+1}{2}\rceil .$$

The conjecture of this problem is that $\nu (d)=2d+1$. This has been proven for $d=2,3,4$.Template:R