In 1972, David G. Larman wrote about the following problem:[1]
Determine the largest number such that for any given points in general position in the -dimensional affine space there is a projective transformation mapping these points into convex position (so they form the vertices of a convex polytope).
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:
Jorge Luis Ramírez Alfonsín proved in 2001 that[3]
The conjecture of this problem is that . This has been proven for .[1][4]
References[]
^ abcLarman, D. G. (1972), "On sets projectively equivalent to the vertices of a convex polytope", The Bulletin of the London Mathematical Society, 4: 6–12, doi:10.1112/blms/4.1.6, MR0307040
^Las Vergnas, Michel (1986), "Hamilton paths in tournaments and a problem of McMullen on projective transformations in ", The Bulletin of the London Mathematical Society, 18 (6): 571–572, doi:10.1112/blms/18.6.571, MR0859948