|Born||11 May 1942|
|Alma mater||Trinity College, Cambridge|
|Known for||Upper bound theorem, McMullen problem|
|Institutions||Western Washington University (1968–1969)|
University College London
Education and careerEdit
McMullen is known for his work in polyhedral combinatorics and discrete geometry, and in particular for proving what was then called the upper bound conjecture and now is the upper bound theorem. This result states that cyclic polytopes have the maximum possible number of faces among all polytopes with a given dimension and number of vertices. McMullen also formulated the g-conjecture, later the g-theorem of Louis Billera, Carl W. Lee, and Richard P. Stanley, characterizing the f-vectors of simplicial spheres.
The McMullen problem is an unsolved question in discrete geometry named after McMullen, concerning the number of points in general position for which a projective transformation into convex position can be guaranteed to exist. It was credited to a private communication from McMullen in a 1972 paper by David G. Larman.
Awards and honoursEdit
- Research papers
- McMullen, P. (1970), "The maximum numbers of faces of a convex polytope", Mathematika, 17 (2): 179–184, doi:10.1112/s0025579300002850, MR 0283691.
- —— (1975), "Non-linear angle-sum relations for polyhedral cones and polytopes", Mathematical Proceedings of the Cambridge Philosophical Society, 78 (2): 247–261, Bibcode:1975MPCPS..78..247M, doi:10.1017/s0305004100051665, MR 0394436.
- —— (1993), "On simple polytopes", Inventiones Mathematicae, 113 (2): 419–444, Bibcode:1993InMat.113..419M, doi:10.1007/BF01244313, MR 1228132, S2CID 122228607.
- Survey articles
- ——; Schneider, Rolf (1983), "Valuations on convex bodies", Convexity and its applications, Basel: Birkhäuser, pp. 170–247, MR 0731112. Updated as "Valuations and dissections" (by McMullen alone) in Handbook of convex geometry (1993), MR1243000.
- ——; Shephard, Geoffrey C. (1971), Convex Polytopes and the Upper Bound Conjecture, Cambridge University Press.
- ——; Schulte, Egon (2002), Abstract regular polytopes, Encyclopedia of Mathematics and its Applications, 92, Cambridge: Cambridge University Press, doi:10.1017/CBO9780511546686, ISBN 0-521-81496-0, MR 1965665.
- Peter McMullen, Peter M. Gruber, retrieved 2013-11-03.
- UCL IRIS information system, accessed 2013-11-03.
- Peter McMullen Collection, 1967-1968, Special Collections, Wilson Library, Western Washington University, retrieved from worldcat.org 2013-11-03.
- Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, 152, Springer, p. 254, ISBN 9780387943657,
Finally, in 1970 McMullen gave a complete proof of the upper-bound conjecture – since then it has been known as the upper bound theorem. McMullen's proof is amazingly simple and elegant, combining to key tools: shellability and h-vectors.
- Gruber, Peter M. (2007), Convex and discrete geometry, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 336, Berlin: Springer, p. 265, ISBN 978-3-540-71132-2, MR 2335496,
The problem of characterizing the f-vectors of convex polytopes is ... far from a solution, but there are important contributions towards it. For simplicial convex polytopes a characterization was proposed by McMullen in the form of his celebrated g-conjecture. The g-conjecture was proved by Billera and Lee and Stanley.
- Larman, D. G. (1972), "On sets projectively equivalent to the vertices of a convex polytope", The Bulletin of the London Mathematical Society, 4: 6–12, doi:10.1112/blms/4.1.6, MR 0307040
- ICM 1974 proceedings Archived 2017-12-04 at the Wayback Machine.
- Awards, Appointments, Elections & Honours, University College London, June 2006, retrieved 2013-11-03.
- List of AMS fellows, retrieved 2013-11-03.