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 iff:


 * $\forall x,y \in A: \forall t \in \left[{0 .. 1}\right]: t x + \left({1 - t}\right) y \in A$.

Line segment
The set


 * $\left\{{ t x + \left({1 - t}\right) y: t \in \left[{0 .. 1}\right] }\right\}$

is often 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.

Examples of Convex Sets

 * Any linear subspace of $V$ is a convex set (proof).
 * Any singleton in $V$ is a convex set (proof)
 * An arbitrary intersection of convex sets is again convex (proof)