Education and career
He 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 Billera, Lee, and Stanley, characterizing the f-vectors of simplicial spheres.
Awards and honours
- Research papers
- McMullen, P. (1970), "The maximum numbers of faces of a convex polytope", Mathematika, 17: 179–184, MR 0283691, doi:10.1112/s0025579300002850.
- —— (1975), "Non-linear angle-sum relations for polyhedral cones and polytopes", Mathematical Proceedings of the Cambridge Philosophical Society, 78 (2): 247–261, MR 0394436, doi:10.1017/s0305004100051665.
- —— (1993), "On simple polytopes", Inventiones Mathematicae, 113 (2): 419–444, MR 1228132, doi:10.1007/BF01244313.
- Survey articles
- ——; Schneider, Rolf (1983), "Valuations on convex bodies", Convexity and its applications, Basel: Birkhäuser, pp. 170–247, MR 731112. Updated as "Valuations and dissections" (by McMullen alone) in Handbook of convex geometry (1993), MR1243000.
- ——; Shephard, G. 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, ISBN 0-521-81496-0, MR 1965665, doi:10.1017/CBO9780511546686.
- 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 onvex 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.
- ICM 1974 proceedings.
- Awards, Appointments, Elections & Honours, University College London, June 2006, retrieved 2013-11-03.
- List of AMS fellows, retrieved 2013-11-03.