Definition:Convex Set (Vector Space)

This page is about Convex Set in the context of Linear Algebra. For other uses, see Convex Set.

Definition

Let $\Bbb F \in \set {\R, \C}$.

Let $V$ be a vector space over $\Bbb F$.

Let $C \subseteq V$.

Definition 1

We say that $C$ is convex if and only if:

$t x + \paren {1 - t} y \in C$

for each $x, y \in C$ and $t \in \closedint 0 1$.

Definition 2

We say that $C$ is convex if and only if:

$t C + \paren {1 - t} C \subseteq C$

for each $t \in \closedint 0 1$, where $t C + \paren {1 - t} C$ denotes a linear combination of subsets.

Line Segment

The set:

$\set {t x + \paren {1 - t} y: t \in \closedint 0 1}$

is called the (straight) line segment joining $x$ and $y$.

A convex set can thus be described as a set containing all straight line segments between its elements.

Also see

• Equivalence of Definitions of Convex Set in Vector Space

• Linear Subspace is Convex Set
• Singleton is Convex Set
• Intersection of Convex Sets is Convex Set

• Definition:Convex Hull

• Results about convex sets can be found here.