Is the Lebesgue measure of the boundary of a union of $n$-balls zero?

Question

I'm working in $\mathbb{R}^n$, but I believe that the question is relevant in any metric space.

Let $B_1^n$ be the closed ball of radius 1 centered at the origin, and let $\oplus$ denote Minkowski addition. For compact $T\subset \mathbb{R}^n$, is it true that the set $$\partial\big( T \oplus B_1^n\big)$$ has 0 Lebesgue measure?

Note that $T\oplus B_1^n = \{t+x: t\in T, x\in B_1^n\}$.

Would it help to consider a countably dense subset of $T$?

I would say yes, since sets with boundary whose boundary have positive lebesgue measure tend to be dusty, or very rough. The boundary of a union of balls should be very smooth. — Ryan, Apr 06 '23 at 08:46
Related: Vitali covering lemma https://en.wikipedia.org/wiki/Vitali_covering_lemma#Infinite_version — GEdgar, Apr 06 '23 at 11:17

score 1 · Answer 1 · answered Apr 06 '23 at 09:03

1

With a litle more research, I noticed that my set $\partial (T\oplus B_1^n)$ is contained in the set $A$ defined in this Stack Exchange question. So, yes, it has Lebesgue measure 0.

answered Apr 06 '23 at 09:03

Ryan

169

Is the Lebesgue measure of the boundary of a union of $n$-balls zero?

1 Answers1