Proving a convex function on an open convex set satisfies some inequalities

[michael.wes]

Homework Statement

Let $$f:\mathcal{O}\subset\mathbb{R}^n\rightarrow\mathbb{R}, \mathcal{O}$$ is an open convex set. Assume that $$D^2f(x)$$ is positive semi-definite $$\forall x\in\mathcal{O}$$. Such $$f$$ are said to be convex functions.

Homework Equations

Prove that $$f((1-t)a+tb)\leq (1-t)f(a)+tf(b),a,b\in\mathcal{O},0 \leq t \leq 1$$ and interpret the result geometrically. (The interpretation is easy, it's the proof of the inequality I'm stuck with)

The Attempt at a Solution

In an earlier part of the question I proved that we have$$f(x)\geq f(a) + \grad f(a)\cdot (x-a) \forall x,a\in\mathcal{O}$$

I have tried to use the mean value theorem in R^n in an attempt to link this with that result and the gradient, but it doesn't help since you lose information in using the mean value theorem, and this isn't an existence result, so it makes me think that there is a direct approach. This is typically given as the definition of a convex function on the web, and not a theorem, so I couldn't find help elsewhere.

Any help appreciated!

Edit: I'm still completely stuck.

 

[michael.wes]

Any help before tomorrow would be appreciated :)