convex set

A set where the line segment between any two of its points stays entirely inside the set.

A set $C\subseteq\mathbb R^n$ is convex if, for every $x,y\in C$ and every $\theta\in[0,1]$ , the point $\theta x + (1-\theta)y$ (a point on the segment between $x$ and $y$ ) is also in $C$ . Geometrically, a convex set has no dents, holes, or disconnected pieces that a straight segment between two of its own points could poke outside of.