Equivalence of Definitions of Convex Hull

Theorem

The following definitions of the concept of Convex Hull are equivalent:

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 $U \subseteq C$.

Proof

Definition $(1)$ iff Definition $(3)$

This is demonstrated in Convex Hull is Smallest Convex Set containing Set.

$\blacksquare$

Definition $(2)$ iff Definition $(3)$

Let $\bigcap C$ denote the intersection of all convex subsets of $V$ containing $U$.

From Intersection of Convex Sets is Convex Set (Vector Spaces), $\bigcap C$ is a convex subset of $V$.

Let $S$ be the smallest convex subset of $V$ containing $U$

By Intersection is Largest Subset:

$\bigcap C = S$

That is, a convex hull by definition $2$ is the same thing as a convex hull by definition $3$.

$\blacksquare$

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): convex hull