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| The '''McMullen problem''' is an open problem in [[discrete geometry]] named after [[Peter McMullen]].
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| ==Statement==
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| In 1972, McMullen has proposed the following problem:<ref name="L">L. G. Larman(1972), "On Sets Projectively Equivalent to the Vertices of a Convex Polytope", ''Bull. London Math. Soc.'' '''4''', pp.6–12</ref>
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| : Determine the largest number <math>\nu(d)</math> such that any given <math>\nu(d)</math> points in [[general position]] in affine ''d''-space '''[[real number|R]]'''<sup>''d''</sup> there is a [[projective transformation]] mapping these points onto the vertices of a [[convex polytope]].
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| ==Equivalent formulations==
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| ===Gale transform===
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| Using the [[Gale transform]], this problem can be reformulate as:
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| : Determine the smallest number <math>\mu(d)</math> such that every set of <math>\mu(d)</math> points ''X'' = {''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''μ''(''d'')</sub>} in linearly general position on '''S'''<sup>d-1</sup> it is possible to choose a set ''Y'' = {''ε''<sub>1</sub>''x''<sub>1</sub>,''ε''<sub>2</sub>''x''<sub>2</sub>,...,''ε''<sub>''μ''(''d'')</sub>''x''<sub>''μ''(''d'')</sub>} where ''ε''<sub>''i''</sub> = ±1 for ''i'' = 1, 2, ..., ''μ''(''d''), such that every open hemisphere of '''S'''<sup>''d''−1</sup> contains at least two members of Y.
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| The number <math>\mu(k)</math>, <math>\nu(d)</math> are connected by the relationships
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| : <math>\mu(k)=\min\{w \mid w\leq\nu(w-k-1)\} \, </math>
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| : <math>\nu(d)=\max\{w \mid w\geq\mu(w-d-1)\} \, </math>
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| ===Partition into nearly-disjoint hulls===
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| Also, by simple geometric observation, it can be reformulate as:
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| : Determine the smallest number <math>\lambda(d)</math> such that for every set ''X'' of <math>\lambda(d)</math> points in '''[[real number|R]]'''<sup>''d''</sup> there exists a [[Partition of a set|partition]] of ''X'' into two sets ''A'' and ''B'' with
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| :: <math>\operatorname{conv}(A\backslash \{x\})\cap \operatorname{conv}(B\backslash \{x\})\not=\varnothing,\forall x\in X. \, </math> | |
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| The relation between <math>\mu</math> and <math>\lambda</math> is
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| : <math>\mu(d+1)=\lambda(d),\qquad d\geq1 \, </math>
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| ===Projective duality===
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| [[File:Pentagon dual arrangement.svg|thumb|300px|An [[arrangement of lines]] dual to the regular pentagon. Every five-line projective arrangement, like this one, has a cell touched by all five lines. However, adding the [[line at infinity]] produces a six-line arrangement with six pentagon faces and ten triangle faces; no face is touched by all of the lines. Therefore, the solution to the McMullen problem for ''d'' = 2 is ''ν'' = 5.]]
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| The equivalent [[projective dual]] statement to the McMullen problem is to determine the largest number <math>\nu(d)</math> such that every set of <math>\nu(d)</math> [[hyperplane]]s 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.
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| ==Results==
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| This problem is still open. However, the bounds of <math>\nu(d)</math> are in the following results:
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| *Larman proved that <math>2d+1\leq\nu(d)\leq(d+1)^2</math>. (1972)<ref name="L" />
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| *[[Michel Las Vergnas]] proved that <math>\nu(d)\leq\frac{(d+1)(d+2)}{2}</math>. (1986)<ref name="LV">[[Michel Las Vergnas|M. Las Vergnas]] (1986), "Hamilton Paths in Tournaments and a Problem McMullen on Projective Transformations in '''R'''<sup>d</sup>", ''Bull. London Math. Soc.'' '''18''', pp.571–572</ref>
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| *Alfonsín proved that <math>\nu(d)\leq2d+\lceil\frac{d+1}{2}\rceil</math>. (2001)<ref name="A">J. L. Ramírez Alfonsín(2001), "Lawrence Oriented Matroids and a Problem of McMullen on Projective Equivalences of Polytopes", ''Europ. J. Combinatorics'' '''22''', pp.723–731</ref>
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| The conjecture of this problem is <math>\nu(d)=2d+1</math>, and it is true for d=2,3,4.<ref name="L" /><ref name="F">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", ''Europ. J. Combinatorics'' '''22''', pp.705–708</ref>
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| ==References==
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| {{reflist}}
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| [[Category:Discrete geometry]]
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| [[Category:Unsolved problems in mathematics]]
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