In 1972, McMullen has proposed the following problem:
Determine the largest number
ν ( d ) such that for any given
ν ( d ) 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).
Using the Gale transform, this problem can be reformulate as:
Determine the smallest number
μ ( d ) such that every set of
μ ( d ) 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 μ ( k ) , ν ( d ) are connected by the relationships
μ ( k ) = min { w ∣ w ≤ ν ( w − k − 1 ) } ν ( d ) = max { w ∣ w ≥ μ ( w − d − 1 ) } Also, by simple geometric observation, it can be reformulate as:
Determine the smallest number
λ ( d ) such that for every set
X of
λ ( d ) points in
Rd there exists a
partition of
X into two sets
A and
B with
The relation between μ and λ is
μ ( d + 1 ) = λ ( d ) , d ≥ 1 The equivalent projective dual statement to the McMullen problem is to determine the largest number ν ( d ) such that every set of ν ( 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.
This problem is still open. However, the bounds of ν ( d ) are in the following results:
David Larman proved that 2 d + 1 ≤ ν ( d ) ≤ ( d + 1 ) 2 . (1972)Michel Las Vergnas proved that ν ( d ) ≤ ( d + 1 ) ( d + 2 ) 2 . (1986)Jorge Luis Ramírez Alfonsín proved that ν ( d ) ≤ 2 d + ⌈ d + 1 2 ⌉ . (2001)The conjecture of this problem is ν ( d ) = 2 d + 1 , and it is true for d=2,3,4.
