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

of ${\displaystyle n>d\geq 2}$ distinct points ${\displaystyle \mathbf {x} (t_{i})}$ with ${\displaystyle t_{1}<t_{2}<\ldots <t_{n}}$ on the moment curve.

The combinatorial structure of this polytope is independent of the points chosen, and the resulting polytope has dimension d and n vertices. Its boundary is a (d − 1)-dimensional simplicial polytope denoted Δ(n,d).

The Gale evenness condition provides a necessary and sufficient condition to determine a facet on a cyclic polytope.

Let ${\displaystyle T:=\{t_{1},t_{2},\ldots ,t_{n}\}}$. Then, a ${\displaystyle d}$-subset ${\displaystyle T_{d}\subseteq T}$ forms a facet of ${\displaystyle C(n,d)}$ if and only if any two elements in ${\displaystyle T\setminus T_{d}}$ are separated by an even number of elements from ${\displaystyle T_{d}}$ in the sequence ${\displaystyle (t_{1},t_{2},\ldots ,t_{n})}$.

Cyclic polytopes are examples of neighborly polytopes, in that every set of at most d/2 vertices forms a face. They were the first neighborly polytopes known, and Theodore Motzkin conjectured that all neighborly polytopes are combinatorially equivalent to cyclic polytopes, but this is now known to be false.