Jordan measurability of the level sets

Question

Let $A$ be a compact subset of $R^n$ and $d_S(\bullet, A)$ be the signed distance function of $A$. Namely, $d_S(p,A) = d\left({p,\partial A} \right)$ for p in A, and $d_S(p,A) = -d\left({p,\partial A} \right)$ for p not in A. Here $d$ denotes the usual Euclidean distance from a point to a set. .

Question 1: Let $r \in R$, and consider the set $A_r$ = {$p : d_S(p,A) \geq r$}. Is the set $A_r$ necessarily Jordan measurable (has boundary of zero Lebesgue measure). If this doesn't hold strictly, then maybe under what conditions on $A$?

Question 2: Let A,B be compact subsets of $R^n$ and $d_S(\bullet, A)$, $d_S(\bullet, B)$ defined as above. For some $t \in R$, consider the set {$p : td_S(p,A) + (1-t)d_S(p,B) \geq 0$}. Is this set Jordan measurable?

Many thanks ahead,

Shay

Answer 1

I was not able to determine if this is a homework problem or not. In any case, here are some observations:

For the first question there are to cases:

1. When $r = 0$ the set $A_r = A$ and the question then is if the boundary of $A$ has Lebesgue measure zero.
2. When $r \ne 0$ the boundary of the set $A_r$ even has finite $n-1$-dimensional Hausdorff measure. This can be seen by considering for each $x \in \partial A_r$ a point $y \in A$ for which $d(x,y) = d(x,A) = |r|$. The open ball $B(y,|r|)$ is contained either in the set $A_r$ (if $r < 0$) or in $\mathbb{R}^n \setminus A_r$ (if $r > 0$). Therefore, starting from each point in $\partial A_r$ you have some cone with a fixed opening angle and radius $|r|$ which does not contain any other points of $\partial A_r$. So, the set $\partial A_r$ is contained in finitely many graphs of Lipschitz functions.

For the second question consider the case $A = B$ and you notice that we are in the situation of the first question with $r=0$.