What does it mean for a function from R^n to R to be convex?

Question

I'm looking to determine when the function $f(\vec{x}) = k\vec{x}\cdot\vec{x} - \vec{x}\cdot\vec{y}$ is convex.

However, I'm not even sure where to start. For a function $\mathbb{R} \rightarrow \mathbb{R}$ I would take the second derivative and set it positive, but I don't know if that still works for functions taking vector input.

Answer 1

Some useful facts:

1. The function $x \mapsto \|x\|^2$ is convex.
2. If you multiply a convex function by a nonnegative number, the resulting function is convex.
3. Any linear function from $\mathbb R^n$ to $\mathbb R$ is convex.
4. The sum of two convex functions is convex.

Using these rules, we can recognize immediately that if $k \geq 0$ then your function is convex.