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