Cyclic polyhedron

A cyclic polyhedron is a convex polyhedron whose vertices lie on a curve $t\mapsto (t,t^{2},\dots ,t^{d})$ ${\ displaystyle t \ mapsto (t, t ^ {2}, \ dots, t ^ {d})}$ $t \ mapsto (t, t ^ {2}, \ dots, t ^ {d})$ at $\mathbb {R} ^{d}$ ${\ displaystyle \ mathbb {R} ^ {d}}$ $\ mathbb {R} ^ {d}$ .

Design

Let be $\mathbf {x} (t)=(t,t^{2},\dots ,t^{d})\in \mathbb {R} ^{d}$ ${\ displaystyle \ mathbf {x} (t) = (t, t ^ {2}, \ dots, t ^ {d}) \ in \ mathbb {R} ^ {d}}$ and $t_{1}<t_{2}<\dots <t_{n}$ ${\ displaystyle t_ {1} <t_ {2} <\ dots <t_ {n}}$ . Convex hull $n$ ${\ displaystyle n}$ points $\mathbf {x} (t_{1}),\mathbf {x} (t_{2}),\ldots ,\mathbf {x} (t_{n})$ ${\ displaystyle \ mathbf {x} (t_ {1}), \ mathbf {x} (t_ {2}), \ ldots, \ mathbf {x} (t_ {n})}$ called $d$ ${\ displaystyle d}$ -dimensional cyclic polyhedron with $n$ ${\ displaystyle n}$ vertices and further denoted $C(n,d)$ ${\ displaystyle C (n, d)}$ .

Properties

Gale Criterion: Let $T=\{t_{1},t_{2},\dots ,t_{n}\}$ ${\ displaystyle T = \ {t_ {1}, t_ {2}, \ dots, t_ {n} \}}$ , and $T_{d}\subset T$ ${\ displaystyle T_ {d} \ subset T}$ - a subset of $d$ ${\ displaystyle d}$ elements. Hyperface in $C(n,d)$ ${\ displaystyle C (n, d)}$ corresponds to $T_{d}$ ${\ displaystyle T_ {d}}$ if and only if between any two adjacent numbers in $T_{d}$ ${\ displaystyle T_ {d}}$ lies an even number of numbers from $T$ ${\ displaystyle T}$ .
Any $⌊$ $d$ $2$ $⌋$ {\ displaystyle \ lfloor {\ tfrac {d} {2}} \ rfloor} peaks in $C$ $($ $n$ $,$ $d$ $)$ {\ displaystyle C (n, d)} form a face.
- In particular, any two vertices of a 4-dimensional cyclic polyhedron are connected by an edge.
Number $i$ {\ displaystyle i} -dimensional faces in $C$ $($ $n$ $,$ $d$ $)$ {\ displaystyle C (n, d)} at $0$ $\leq$ $i$ $<$ $⌊$ $d$ $2$ $⌋$ {\ displaystyle 0 \ leq i <\ lfloor {\ frac {d} {2}} \ rfloor} equally $($ $n$ $i$ $+$ $one$ $)$ {\ displaystyle {\ binom {n} {i + 1}}} .
- Using the Dehn - Somerville identities , one can find the number of faces of higher dimensions.
- For anyone $k$ ${\ displaystyle k}$ among all $d$ ${\ displaystyle d}$ -dimensional polyhedra with $n$ ${\ displaystyle n}$ vertices cyclic polyhedra have a maximum number $k$ ${\ displaystyle k}$ -dimensional faces.

Literature

V.A. Timorin. Combinatorics of convex polyhedra . - MCCMO, 2002. - (Summer School "Contemporary Mathematics"). - ISBN 5-94057-024-0 .

Cyclic polyhedron

Design

Properties

Literature

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