Polytope

Definition

Let $n \ge 1$ . A polytope in $\R^n$ is a set $P = P_1 \cup \cdots \cup P_k$ that is the union of finitely many convex polytopes $P_1, \ldots, P_k$ in $\R^n$ .

Remarks

Example: the L-shaped region $[0, 2] \times [0, 1] \cup [0, 1] \times [0, 2]$ is the union of two rectangles, which are convex polytopes, so it is a polytope. It is not convex: it contains $(2, 1)$ and $(1, 2)$ , but not their midpoint $(3/2, 3/2)$ . Non-example: the closed unit disc in $\R^2$ is not a polytope. The boundary of a finite union of convex polygons lies in finitely many lines. Each line meets the unit circle in at most two points, so this boundary cannot contain the whole circle. Conventions differ between authors. In convex geometry and in polyhedral combinatorics "polytope" usually means convex polytope, and some authors require a polytope to be a union of simplices that meet in common faces.

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Convex polytope concept
Polytope definition

Tag	`ET5G9CT0RC`
URI	`m2://ET5G9CT0RC`
Kind	definition
Writing	W0 stub
MSC	52B11, 52B05
Defines	Polytope
Uses	Convex polytope

Polytope

Definition

Remarks

Reading path

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