Examining the function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ with $f(x_1,x_2) = 4x_1^2+x_2^2$ for uniform convexity.

Question

A quick question from a multiple choice test I am preparing for:

Is the function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ with $f(x_1,x_2) = 4x_1^2+x_2^2$ uniformly convex?

Answer 1

The key is to remember the fact that

$f$ is uniformly convex (on $X$) iff there is a $\mu > 0$ such that $$d^T \nabla^2(y)d \ge \mu \|d\|^2$$ for all $x \in X$ and all $d \in \mathbb{R}^n$.

This is easy to check.