Cyclic polytope

A cyclic polytope is a convex polytope with corners on the moment curve . It is of great importance for many questions in the combinatorial theory of polytopes, including the upper-bound theorem .

definition

Let the moment curve be in dimension d. ${\displaystyle x(t)=(t,t^{2},\dotsc ,t^{d})\in \mathbb {R} ^{d}}$

Then the cyclic polytope is the convex hull of n points on the moment curve, where n is not smaller than d + 1. ${\displaystyle C(d,n)}$

${\displaystyle C(d,n)=\mathrm {conv} ({x(t_{1}),x(t_{2}),\dotsc ,x(t_{n})})}$ With ${\displaystyle t_{1}<t_{2}<\cdots <t_{n}}$

It is also possible to define the cyclic polytope on differently defined moment curves.

Examples

In two-dimensional the moment curve is identical to the normal parabola . Every polygon whose corners lie on the normal parabola is a cyclic polytope.

properties

• Two cyclic polytopes of equal dimensions with the same number of corners are combinatorially equivalent . So one can speak of the cyclic d-polytope with n vertices. This property follows from Gale's straightness criterion .
• The cyclic polytope is a simplicial polytope, i. H. each of its real sides is a simplex .
• Furthermore is a - neighborly polytope. Every convex hull of any set of vertices is one side of the polytope. ${\displaystyle C(d,n)}$ ${\displaystyle \lfloor d/2\rfloor }$ ${\displaystyle \lfloor d/2\rfloor }$
• The most outstanding property of the cyclic polytope is its "extremity". Among all d-dimensional polytopes with n vertices, the maximum number of k-dimensional sides has (k <d). A d-polytope with n corners can therefore not have more k-sides than the corresponding cyclic polytope with n corners ( upper bound theorem ). ${\displaystyle C(d,n)}$

literature

• Branko Grünbaum with Volker Kaibel, Victor Klee, Günter M. Ziegler : Convex Polytopes. 2nd edition, Springer 2003, ISBN 0-387-00424-6 (first from Grünbaum 1967).