Cyclic polytope

Gale evenness condition

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

Let ${\displaystyle T:=\{t_1,t_2,\dots ,t_n\}}$. 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 ${\displaystyle (t_1,t_2,\dots ,t_n)}$.

Neighborliness

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.

Number of faces

The number of i-dimensional faces of the cyclic polytope Δ(n,d) is given by the formula

${\displaystyle f_i(\mathrm{\Delta }(n,d))=(\genfrac{}{}{0ex}{}{n}{i+1})\text{for}0\le i<\lfloor \frac{d}{2}\rfloor }$

and ${\displaystyle (f_0,\dots ,f_{\lfloor \frac{d}{2}\rfloor -1})}$ completely determine ${\displaystyle (f_{\lfloor \frac{d}{2}\rfloor },\dots ,f_{d-1})}$ via the Dehn–Sommerville equations.

Upper bound theorem

The upper bound theorem states that cyclic polytopes have the maximum possible number of faces for a given dimension and number of vertices: if Δ is a simplicial sphere of dimension d − 1 with n vertices, then

${\displaystyle f_i(\mathrm{\Delta })\le f_i(\mathrm{\Delta }(n,d))\text{for}i=0,1,\dots ,d-1.}$

The upper bound conjecture for simplicial polytopes was proposed by Theodore Motzkin in 1957 and proved by Peter McMullen in 1970. Victor Klee suggested that the same statement should hold for all simplicial spheres and this was indeed established in 1975 by Richard P. Stanley using the notion of a Stanley–Reisner ring and homological methods.