McMullen problem


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

Statement

In 1972, McMullen proposed the following problem:

Equivalent formulations

Gale transform

Using the Gale transform, this problem can be reformulated as:
The number, are connected by the relationships

Partition into nearly-disjoint hulls

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

Projective duality

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:
The conjecture of this problem is, and it is true for d = 2, 3, 4.