Definition:Convex Hull
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Definition
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 $U \subseteq C$.
Also see
- Results about convex hulls can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hull, convex
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hull, convex