Definition:Convex Hull

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

• Equivalence of Definitions of Convex Hull

• Convex Hull is Smallest Convex Set containing Set
• Definition:Closed Convex Hull

• 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