Cyclic polytope

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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 Rd, 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 fi of i-dimensional faces among all simplicial spheres of dimension d − 1 with n vertices.

Definition[edit]

The moment curve in is defined by

.[1]

The -dimensional cyclic polytope with vertices is the convex hull

of distinct points with on the moment curve.[1]

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

Gale evenness condition[edit]

The Gale evenness condition[2] provides a necessary and sufficient condition to determine a facet on a cyclic polytope.

Let . Then, a -subset forms a facet of iff any two elements in are separated by an even number of elements from in the sequence .

Neighborliness[edit]

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.[3][4]

Number of faces[edit]

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

and completely determine via the Dehn–Sommerville equations.

Upper bound theorem[edit]

Main article: 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

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[5] using the notion of a Stanley–Reisner ring and homological methods.

See also[edit]

References[edit]

  1. ^ a b c Miller, Ezra; Sturmfels, Bernd (2005). Combinatorial commutative algebra. Graduate Texts in Mathematics. 227. New York, NY: Springer-Verlag. p. 119. ISBN 0-387-23707-0. Zbl 1090.13001. 
  2. ^ Ziegler, Günter (1994). Lectures on Polytopes. Springer. p. 14. ISBN 0-387-94365-X. 
  3. ^ Gale, David (1963), "Neighborly and cyclic polytopes", in Klee, Victor, Convexity, Seattle, 1961, Symposia in Pure Mathematics, 7, American Mathematical Society, pp. 225–233, ISBN 978-0-8218-1407-9 .
  4. ^ Shermer, Ido (1982), "Neighborly polytopes", Israel Journal of Mathematics, 43 (4): 291–311, doi:10.1007/BF02761235 .
  5. ^ Stanley, Richard (1996). Combinatorics and commutative algebra. Boston, MA: Birkhäuser Boston, Inc. p. 164. ISBN 0-8176-3836-9.