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$ .

What is the most efficient way to test whether a set $X \subset \{0, 1\}^n$ and its complement $\{0, 1\}^n \setminus X$ are linearly separable?

I am interested in algorithms that have optimal running time, and ideally which are also very easy to implement. If you can also give some tips on how to implement the algorithm(s) you mention in the answer, that would be great.