For how many points is it always possible to projectively transform the points into convex position?
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 and is
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.
This problem is still open. However, the bounds of are in the following results:
The conjecture of this problem is that . This has been proven for .[1][4]