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.

We say $f$ is convex if given any two $x,y\in \mathbb{R}^{n}$ and any $\lambda \in [0,1]$, we have $f(\lambda x +(1-\lambda) y)\le \lambda f(x)+ (1-\lambda)f(y)$ — Ben, Apr 25 '19 at 02:27
There is also a sufficient condition for convexity involving second derivatives, see the first point here: https://en.wikipedia.org/wiki/Convex_function#Functions_of_several_variables — Theo Bendit, Apr 25 '19 at 02:55
@littleO That's exactly what I needed; if you write that as an answer I'll upvote and accept! — Draconis, Apr 25 '19 at 03:27

score 0 · Answer 1 · answered Apr 25 '19 at 08:32

Some useful facts:

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

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

1 Answers1