Cyclic polytope

Builds on Convex polytope Convex hull Facet

Definition

Let $d \ge 2$ and $n > d$ be integers. The moment curve in $\R^d$ is the map $x \colon \R \to \R^d$ , $x(t) = (t, t^2, \dots, t^d)$ . Let $t_1 < t_2 < \cdots < t_n$ be real numbers. The cyclic polytope $C(n, d)$ is the convex hull of the points $x(t_1), \dots, x(t_n)$ .

Remarks

$C(n,d)$ is a $d$ -dimensional convex polytope whose vertices are all $n$ points $x(t_i)$ . Its face lattice does not depend on the choice of $t_1 < \cdots < t_n$ , so $C(n, d)$ names a well-defined combinatorial type. Gale's evenness condition describes its facets. A set $S$ of $d$ indices gives a facet if and only if, for any two indices $i < j$ outside $S$ , an even number of elements of $S$ lie between them.

Example: for $d = 2$ the moment curve is the parabola $y = x^2$ , and $C(n, 2)$ is a convex $n$ -gon. Its facets are the edges $\set{i, i+1}$ and $\set{1, n}$ , and these are exactly the pairs passing the evenness condition.

Non-example: the regular octahedron is not of the form $C(6, 3)$ . In $C(6,3)$ the vertex $x(t_1)$ lies in $5$ edges, while every vertex of the octahedron lies in $4$ edges.

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Convex hull concept
Convex polytope concept
Facet concept
Cyclic polytope definition

Tag	`TQ2TZDMAPA`
URI	`m2://TQ2TZDMAPA`
Kind	definition
Writing	W0 stub
MSC	52B12, 52B05, 52B11
Defines	Cyclic polytope
Uses	Convex polytope, Convex hull, Facet

Cyclic polytope

Definition

Remarks

Reading path

Further reading

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