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 isfor 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.