We know that a function is convex if we have $$\lambda f(x_1) + (1-\lambda)f(x_2) \ge f(\lambda x_1 + (1-\lambda)x_2)$$ where $0\le\lambda\le1$
But I don't know where is it come from ? Unfortunately , I can't understand it. I searched in the internet many times but it didn't help to me. If someone explain this expression is helpful.
The given inequality means that whenever you take two points on the graph of $f$, then the segment joining them is above the graph itself.
Edit. For the proof, it suffices to notice that for all $A,B\in\mathbb{R}^2$, one has: $$[AB]=\{(1-t)A+tB;t\in[0,1]\}.$$ This is essentially the definition of $[AB]$, which is the set of barycentric combinations of $A$ and $B$.