convex

convex set

check the definition in real analysis here convex set

Set $S\subseteq {\mathbb{R}}^n$ is a convex set if it contains a line segment between any two of it’s points.

convex function

Function $f:S\subseteq {\mathbb{R}}^n⟶\mathbb{R}$ is convex function on convex set $S$ if it’s graph along any line segment in $S$ lies on or below chord connecting function values at endpoints of segment.

Def

Any local minimum of convex function $f$ on convex set $S$ is global minimum of $f$ on $S$.

Def

Any local minimum of strictly convex function $f$ on convex set $S$ is unique global minimum of $f$ on $S$.