The McMullen problem is an open problem in

discrete geometry Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic ge ...

named after

Peter McMullen Peter McMullen (born 11 May 1942) is a British mathematician, a professor emeritus of mathematics at University College London. Education and career McMullen earned bachelor's and master's degrees from Trinity College, Cambridge, and studied at ...

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 In the mathematical discipline of polyhedral combinatorics, the Gale transform turns the vertices of any convex polytope into a set of vectors or points in a space of a different dimension, the Gale diagram of the polytope. It can be used to descr ...

, this problem can be reformulated as: The numbers

\nu

of the original formulation of the McMullen problem and

\mu

of the Gale transform formulation are connected by the relationships

\begin
\mu(k)&=\min\ \\
\nu(d)&=\max\ 
\end

Partition into nearly-disjoint hulls

Also, by simple geometric observation, it can be reformulated as: The relation between

\mu

and

\lambda

\mu(d+1)=\lambda(d),\qquad  d\geq1 \,

Projective duality

The equivalent projective dual statement to the McMullen problem is to determine the largest number

\nu(d)

such that every set of

\nu(d)

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hype ...

s in general position in ''d''-dimensional

real projective space In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space. Basic properties Construction ...

form an

arrangement of hyperplanes In geometry and combinatorics, an arrangement of hyperplanes is an arrangement of a finite set ''A'' of hyperplanes in a linear, affine, or projective space ''S''. Questions about a hyperplane arrangement ''A'' generally concern geometrical, ...

in which one of the cells is bounded by all of the hyperplanes.

Results

This problem is still open. However, the bounds of

\nu(d)

are in the following results: *David Larman proved in 1972 that

2d+1\leq\nu(d)\leq(d+1)^2.

* Michel Las Vergnas proved in 1986 that

\nu(d)\leq\frac.

*Jorge Luis Ramírez Alfonsín proved in 2001 that

\nu(d)\leq2d+\left\lceil\frac\right\rceil.

The conjecture of this problem is that

\nu(d)=2d+1

. This has been proven for

d=2,3,4

References

{{reflist, refs= {{citation , last = Ramírez Alfonsín , first = J. L. , doi = 10.1006/eujc.2000.0492 , issue = 5 , journal =

European Journal of Combinatorics European, or Europeans, or Europeneans, may refer to: In general * ''European'', an adjective referring to something of, from, or related to Europe ** Ethnic groups in Europe ** Demographics of Europe ** European cuisine European cuisine co ...

, mr = 1845496 , pages = 723–731 , title = Lawrence oriented matroids and a problem of McMullen on projective equivalences of polytopes , volume = 22 , year = 2001, doi-access = free {{citation , last1 = Forge , first1 = David , last2 = Las Vergnas , first2 = Michel , author2-link = Michel Las Vergnas , last3 = Schuchert , first3 = Peter , department = Combinatorial geometries (Luminy, 1999) , doi = 10.1006/eujc.2000.0490 , issue = 5 , journal =

, mr = 1845494 , pages = 705–708 , title = 10 points in dimension 4 not projectively equivalent to the vertices of a convex polytope , volume = 22 , year = 2001, doi-access = free {{citation , last = Larman , first = D. G. , doi = 10.1112/blms/4.1.6 , journal =

The Bulletin of the London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical ...

, mr = 307040 , pages = 6–12 , title = On sets projectively equivalent to the vertices of a convex polytope , volume = 4 , year = 1972 {{citation , last = Las Vergnas , first = Michel , author-link = Michel Las Vergnas , doi = 10.1112/blms/18.6.571 , issue = 6 , journal =

, mr = 859948 , pages = 571–572 , title = Hamilton paths in tournaments and a problem of McMullen on projective transformations in

\R^d

, volume = 18 , year = 1986 Discrete geometry Unsolved problems in geometry