cyclic polytope について

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cyclic polytope ：ウィキペディア英語版

cyclic polytope
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.
== Definition ==
The moment curve in

\mathbb^d

is defined by
:

\mathbf:\mathbb \rightarrow \mathbb^d, \mathbf(t) := \begint,t^2,\ldots,t^d\end^T

.
The

d

-dimensional cyclic polytope with

n

vertices is the convex hull
:

C(n,d) := \mathbf\(t_2),\ldots,\mathbf(t_n) \}

n > d \ge 2

distinct points

\mathbf(t_i)

with

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'').

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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