Let $ A$ be a measurable subset of the metric space $ \mathcal X = ([0, 1]^n,\ell_p)$ , and define its $ \epsilon$ blowup by $ A^\varepsilon:=\{x \in \mathcal X \mid \xa\_p \le \epsilon\text{ for some }a \in A\}$ .
Question

If $ \operatorname{vol}(A) > 0$ , what is a good lower bound on $ \operatorname{vol}(A^\epsilon)$ ?

Same question with $ \operatorname{vol}(A) \ge 1/2$ .