Definition:Convex Hull/Definition 1

Definition

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

Let $U \subseteq V$.

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

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

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That is, it is the set of all convex combinations of elements of $U$.

Also see

• Equivalence of Definitions of Convex Hull

• Results about convex hulls can be found here.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): convex hull
• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $21.5$: The Convex Hull