Convex Analysis/Convex sets and convex functions

Definition ():

Let $V$ be a vector space and $f:V\to [-\infty ,\infty ]$ be a function. $f$ is called convex if and only if for all $v,w\in V$ and $t\in [0,1]$ the following identity holds:

$f(tv+(1-t)w)\leq tf(v)+(1-t)f(v)$

Intuitively, this means if two points $v,w\in V$ are given, then the line connecting $(v,f(v))$ to $(w,f(w))$ lies above the path $t\mapsto (tv+(1-t)w,f(tv+(1-t)w))$.