Convex polytope

Definition

Let $n \ge 1$ . A convex polytope in $\R^n$ is a set of the form $P = \mathrm{conv}\set{v_1, \ldots, v_m} = \set{\lambda_1 v_1 + \cdots + \lambda_m v_m : \lambda_i \ge 0, \ \lambda_1 + \cdots + \lambda_m = 1},$ the convex hull of finitely many points $v_1, \ldots, v_m$ of the Euclidean space $\R^n$ . The dimension of $P$ is the dimension of its affine hull.

Remarks

Example: the square $[0, 1]^2$ is $\mathrm{conv}\set{(0, 0), (1, 0), (0, 1), (1, 1)}$ , since every point $(x, y)$ of the square is $(1 - x)(1 - y) (0, 0) + x (1 - y) (1, 0) + (1 - x) y (0, 1) + x y (1, 1)$ , with nonnegative weights of sum $1$ . Non-example: the closed unit disc is compact and convex but not a convex polytope. The convex hull of finitely many points has only finitely many extreme points, while every point of the unit circle is an extreme point of the disc. Equivalently, a convex polytope is a bounded set of the form $\set{x \in \R^n : A x \le b}$ for a real $m \times n$ matrix $A$ and $b \in \R^m$ ; this equivalence is the Minkowski–Weyl theorem. Many authors allow the empty set as a convex polytope, the hull of no points.

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Convex hull concept
Euclidean space object
Convex polytope definition

Tag	`XNTJTH6ACN`
URI	`m2://XNTJTH6ACN`
Kind	definition
Writing	W0 stub
MSC	52B11
Defines	Convex polytope
Uses	Convex hull, Euclidean space

Convex polytope

Definition

Remarks

Reading path

Further reading

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