Category:Definitions/Convex Hulls

This category contains definitions related to convex hulls.
Related results can be found in Category:Convex Hulls.

Let $V$ be a vector space over $\R$.

Let $U \subseteq V$.

Definition 1

The convex hull of $U$ is defined and denoted:

$\ds \map {\operatorname {conv} } U = \set {\sum_{j \mathop = 1}^n \lambda_j u_j : n \in \N, \, u_j \in U \text { and } \lambda_j \in \R_{> 0} \text { for each } j, \, \sum_{j \mathop = 1}^n \lambda_j = 1}$

Definition 2

The convex hull of $U$ is defined and denoted:

$\ds \map {\operatorname {conv} } U = $ the intersection of all convex sets $C \subseteq V$ of $V$ such that $U \subseteq C$.

Definition 3

The convex hull of $U$ is defined and denoted:

$\ds \map {\operatorname {conv} } U = $ the smallest convex set $C$ such that $U \subseteq C$.