# Definition:Convex Set (Vector Space)

## Definition

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

A subset $A \subseteq V$ is said to be convex if and only if:

$\forall x, y \in A: \forall t \in \closedint 0 1: t x + \paren {1 - t} y \in A$

### 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.