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, and strengthened from polytopes to subdivisions of a sphere in 1975 by Richard P. Stanley.
The cyclic polytope Δ(n,d) may be defined as the convex hull of n vertices on the moment curve (t, t2, t3, ...). The precise choice of which n points on this curve are selected is irrelevant for the combinatorial structure of this polytope. The number of i-dimensional faces of Δ(n,d) is given by the formula
and completely determine via the Dehn–Sommerville equations. The same formula for the number of faces holds more generally for any neighborly polytope.
![(f_{0},\ldots ,f_{{[{\frac {d}{2}}]-1}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/31c743adde82d9e182fabb1265b64cbfb0a52f97)
![(f_{{[{\frac {d}{2}}]}},\ldots ,f_{{d-1}})](https://wikimedia.org/api/rest_v1/media/math/render/svg/25e1c34e81c598a528d4a747858607e59519ce20)
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