McMullen problem
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The McMullen problem is an open problem in discrete geometry named after Peter McMullen.
Statement
In 1972, McMullen has proposed the following problem:[1]
- Determine the largest number such that for any given points in general position in affine d-space Rd there is a projective transformation mapping these points into convex position (so they form the vertices of a convex polytope).
Equivalent formulations
Gale transform
Using the Gale transform, this problem can be reformulate as:
- Determine the smallest number such that every set of points X = {x1, x2, ..., xμ(d)} in linearly general position on Sd-1 it is possible to choose a set Y = {ε1x1,ε2x2,...,εμ(d)xμ(d)} where εi = ±1 for i = 1, 2, ..., μ(d), such that every open hemisphere of Sd−1 contains at least two members of Y.
The number , are connected by the relationships
Partition into nearly-disjoint hulls
Also, by simple geometric observation, it can be reformulate as:
- Determine the smallest number such that for every set X of points in Rd there exists a partition of X into two sets A and B with
The relation between and is
Projective duality
![](../I/m/Pentagon_dual_arrangement.svg.png)
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 that . (1972)[1]
- Michel Las Vergnas proved that . (1986)[2]
- Jorge Luis Ramírez Alfonsín proved that . (2001)[3]
The conjecture of this problem is , and it is true for d=2,3,4.[1][4]
References
- 1 2 3 D. G. Larman (1972), "On Sets Projectively Equivalent to the Vertices of a Convex Polytope", Bulletin of the London Mathematical Society 4, pp.6–12
- ↑ M. Las Vergnas (1986), "Hamilton Paths in Tournaments and a Problem McMullen on Projective Transformations in Rd", Bulletin of the London Mathematical Society 18, pp.571–572
- ↑ J. L. Ramírez Alfonsín (2001), "Lawrence Oriented Matroids and a Problem of McMullen on Projective Equivalences of Polytopes", European Journal of Combinatorics 22, pp.723–731
- ↑ D. Forge, M. Las Vergnas and P. Schuchert (2001), "A Set of 10 Points in Dimension 4 not Projectively Equivalent to the Vertices of Any Convex Polytope", European Journal of Combinatorics 22, pp.705–708