# Convex Analysis/Convex sets and convex functions

Definition (Let ${\displaystyle V}$ be a vector space and ${\displaystyle f:V\to [-\infty ,\infty ]}$ be a function. ${\displaystyle f}$ is called convex if and only if for all ${\displaystyle v,w\in V}$ and ${\displaystyle t\in [0,1]}$ the following identity holds:
${\displaystyle f(tv+(1-t)w)\leq tf(v)+(1-t)f(v)}$):
Intuitively, this means if two points ${\displaystyle v,w\in V}$ are given, then the line connecting ${\displaystyle (v,f(v))}$ to ${\displaystyle (w,f(w))}$ lies above the path ${\displaystyle t\mapsto (tv+(1-t)w,f(tv+(1-t)w))}$.