= Hit-and-run kernel density
For a planar <hit-and-run sampler>, let $\ell(x,u)$ be the chord length and $a(x)$ the fraction of directions with positive chord length. With zero-length chords redrawn, the <probability density function> is
$$
p(x,y)=\frac1{\pi a(x)\ell(x,(y-x)/|y-x|)|y-x|}\quad(y\ne x)
$$
for almost every $y$ in the body. Interior points have $a(x)=1$. The two signed representations of a direction contribute the factor two in the polar-coordinate calculation. If the body has diameter $D$, a version of this density is bounded below by $1/(\pi D^2)$. This is a global <Doeblin condition> after normalization by the body's area.
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