# Isoperimetry on $[0, 1]^n$ w.r.t $\ell_p$ distance

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

# Question

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

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