Why does convexity of a function required the following

Question

What is the significance of the following condition

$$\forall x_1, x_2 \in dom(f) , \forall \theta \in [0, 1], f(\theta x - (1-\theta)y) \leq \theta f(x) + (1-\theta)f(y)$$

and why isn't the condition simply

$$range(f) \text{ is convex} $$

Answer 1

The difference is that one condition is algebraic and one is geometric. both are needed to develop intuition of both kinds.

The correct geometric condition, by the way, is that the epigraph of $f$ be convex, not the range. Consider, e.g. $f(x) = -|x|$ with a perfectly convex range $(-\infty,0]$ (but the function, of course, is concave, not convex).

EDIT As for the meaning behind the algebraic intuition, if you assume $f''$ exists, then the convexity condition (your inequality in the question) is equivalent to $f''(x) \ge 0$. The proof can be found on ProofWiki (see the Corollary there).