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:
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 ν {\displaystyle \nu } of the original formulation of the McMullen problem and μ {\displaystyle \mu } of the Gale transform formulation are connected by the relationships
μ ( k ) = min { w ∣ w ≤ ν ( w − k − 1 ) } ν ( d ) = max { w ∣ w ≥ μ ( w − d − 1 ) } {\displaystyle {\begin{aligned}\mu (k)&=\min\{w\mid w\leq \nu (w-k-1)\}\\\nu (d)&=\max\{w\mid w\geq \mu (w-d-1)\}\end{aligned}}}
Partition into nearly-disjoint hulls Also, by simple geometric observation, it can be reformulated as:
The relation between μ {\displaystyle \mu } and λ {\displaystyle \lambda } is
μ ( d + 1 ) = λ ( d ) , d ≥ 1 {\displaystyle \mu (d+1)=\lambda (d),\qquad d\geq 1\,}
Projective duality
The equivalent projective dual statement to the McMullen problem is to determine the largest number ν ( d ) {\displaystyle \nu (d)} such that every set of ν ( d ) {\displaystyle \nu (d)} 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 ν ( d ) {\displaystyle \nu (d)} are in the following results:
David Larman proved in 1972 that 2 d + 1 ≤ ν ( d ) ≤ ( d + 1 ) 2 . {\displaystyle 2d+1\leq \nu (d)\leq (d+1)^{2}.}
Michel Las Vergnas proved in 1986 that ν ( d ) ≤ ( d + 1 ) ( d + 2 ) 2 . {\displaystyle \nu (d)\leq {\frac {(d+1)(d+2)}{2}}.}
Jorge Luis Ramírez Alfonsín proved in 2001 that ν ( d ) ≤ 2 d + ⌈ d + 1 2 ⌉ . {\displaystyle \nu (d)\leq 2d+\left\lceil {\frac {d+1}{2}}\right\rceil .}
The conjecture of this problem is that ν ( d ) = 2 d + 1 {\displaystyle \nu (d)=2d+1} . This has been proven for d = 2 , 3 , 4 {\displaystyle d=2,3,4} .
References
