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



Jordan measurability of the level sets
*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 $r=0$.


