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.
Statement
In 1972, David G. Larman wrote about the following problem:[1]
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:
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:
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:
- David Larman proved in 1972 that[1]
- Michel Las Vergnas proved in 1986 that[2]
- Jorge Luis Ramírez Alfonsín proved in 2001 that[3]
The conjecture of this problem is that . This has been proven for .[1][4]
References
- 1 2 3 Larman, 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, MR 0307040
- ↑ 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, MR 0859948
- ↑ Ramírez Alfonsín, J. L. (2001), "Lawrence oriented matroids and a problem of McMullen on projective equivalences of polytopes", European Journal of Combinatorics, 22 (5): 723–731, doi:10.1006/eujc.2000.0492, MR 1845496
- ↑ Forge, David; Las Vergnas, Michel; Schuchert, Peter (2001), "10 points in dimension 4 not projectively equivalent to the vertices of a convex polytope", Combinatorial geometries (Luminy, 1999), European Journal of Combinatorics, 22 (5): 705–708, doi:10.1006/eujc.2000.0490, MR 1845494