Let $A\subset \mathbb R^n$ be a compact set, and denote by $\lambda$ the Lebesgue measure on $\mathbb R^n$. Given $\delta>0$ define $$A^\delta := \{x\in\mathbb R^n; |x-a|<\delta\ \text{for some }a\in A\}.$$
Question: Is it possible to find $A$ such that $\lambda(\partial A^\delta) >0$ for every $\delta>0$ small enough?
I believe the above question is false because $A^\delta$ seems to "regularize" the boundary of $A$. However, I could not prove it. Does anyone have any ideas/references?