Upper bound theorem

In mathematics, the upper bound theorem states that cyclic polytopes have the largest possible number of faces among all convex polytopes with a given dimension and number of vertices. It is one of the central results of polyhedral combinatorics.

Originally known as the upper bound conjecture, this statement was formulated by Theodore Motzkin, proved in 1970 by Peter McMullen,^[1] and strengthened from polytopes to subdivisions of a sphere in 1975 by Richard P. Stanley.

^ Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, vol. 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 two key tools: shellability and h-vectors.

[1] Ziegler, Günter M. (1995), Lectures on Polytopes, Graduate Texts in Mathematics, vol. 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 two key tools: shellability and h-vectors.

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