Definition:Convex Set (Vector Space)

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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 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