Let $A$ be a compact subset of $R^n$ and $d_S(\bullet, A)$ be the singed 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

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