Given a bounded domain $\Omega\subset\mathbb R^n$ ($n\geq2$), how often can a single real number $r>0$ appear as a distance of two points on $\partial\Omega$? We can make any assumptions about the boundary $\partial\Omega$ if needed; I would prefer weak assumptions like finite $(n-1)$-dimensional Hausdorff measure, but smoothness is also ok. Specifically, I would be interested in a reference or a proof for either of the following two claims, which intuitively seem true:

Claim 1:Fix $r>0$ and a bounded domain $\Omega\subset\mathbb R^n$ with sufficiently regular boundary. For $x\in\mathbb R^n$ and $v\in S^{n-1}$, denote the line through $x$ in direction $v$ by $L_{x,v}=x+v\mathbb R$. Then for almost every $(x,v)\in\mathbb R^n\times S^{n-1}$ the set $L_{x,v}\cap\partial\Omega$ does not contain any two points exactly distance $r$ from each other.

Claim 2:Fix $r>0$ and a bounded domain $\Omega\subset\mathbb R^n$ with sufficiently regular boundary. Then for almost all $x,y\in\partial\Omega$ we have $d(x,y)\neq r$. (The measure on $\partial\Omega\times\partial\Omega$ is the $2(n-1)$-dimensional Hausdorff measure.)

Both claims are of the form "a set $A$ is of zero measure". (The set $A$ is different in different claims, a subset of $\mathbb R^n\times S^{n-1}$ or $\partial\Omega\times\partial\Omega$.) The set $A$ need not be discrete. For example, if $n=2$ and $\Omega=\{x;x_1>0,x_2>0,|x|<1\}$, the set $A$ is quite large for $r=1$ but both claims are still true.

Distinct Distances Problemis relevant here? Posed in 1946, for $n$ points in $\mathbb{R}^2$, the lowerbound (and perhaps the "right answer") is $\Omega(ne^{\frac{c \log n}{\log \log n}})$. Cf.Research Problems in Discrete Geometry(Springer link). $\endgroup$ – Joseph O'Rourke Feb 24 '15 at 23:59