Maximal difference between two positive semi-definite functions

Question

Given two continuous and positive semi-definite functions $f,g: \mathbb{R}^n \mapsto \mathbb{R}$ is it possible to find the maximal distance between the functions over a bounded subset of $\mathbb{R}^n$ (i.e., $\max\limits_{\vec{x} \in D} \left|f(\vec{x}) - g(\vec{x})\right|$ where $\forall x \in D \subset \mathbb{R}^n: x_L \leq x \leq x_U$)? If not, is it possible to find a (tight) over-approximation?

Answer 1

As a counterexample, consider $g(\vec x)=\alpha f(\vec x)$ for some $\alpha\geq 0$ and $f(\vec x)=|\vec x|$, then you see that there is no upper bound on that max. In fact $$ \max_{\vec x}|f(\vec x)-g(\vec x)|=\alpha\max_{\vec x}|\vec x| $$ which is not bounded from above.

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