Cyclic polytope

In mathematics, a cyclic polytope, denoted C(n,d), is a convex polytope formed as a convex hull of n distinct points on a rational normal curve in R^d, where n is greater than d. These polytopes were studied by Constantin Carathéodory, David Gale, Theodore Motzkin, Victor Klee, and others. They play an important role in polyhedral combinatorics: according to the upper bound theorem, proved by Peter McMullen and Richard Stanley, the boundary Δ(n,d) of the cyclic polytope C(n,d) maximizes the number f_i of i-dimensional faces among all simplicial spheres of dimension d − 1 with n vertices.

The moment curve in ${\displaystyle \mathbb {R} ^{d}}$ is defined by

The ${\displaystyle d}$-dimensional cyclic polytope with ${\displaystyle n}$ vertices is the convex hull

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