Cyclic polytope

Summary

A cyclic polytope is the convex hull of $n \ge d + 1$ distinct points on the moment curve $t \mapsto (t, t^2, \dots, t^d)$ in $\R^d$. Its combinatorial type depends only on the number $n$ of points and the dimension $d$, and it is denoted $C(n, d)$. Cyclic polytopes are simplicial and neighborly. They have the largest number of faces of each dimension among all $d$-polytopes with $n$ vertices, which is the content of the upper bound theorem.