Absolutely continuous curves in spaces of compactly supported densities

We give a constructive proof for existence of absolutely continuous curves connecting each pair $\mu ,\nu \in {\mathrm{P}\mathrm{L}}_{\mathrm{\infty }}^p({\mathbb{R}}^n)$, for every $1\le p\le \mathrm{\infty }$, where $({\mathrm{P}\mathrm{L}}_{\mathrm{\infty }}^p({\mathbb{R}}^n),{\mathfrak{d}}_{\mathrm{\infty }}^p)$ is the complete metric space of absolutely continuous measures with density in $L^p({\mathbb{R}}^n)$ and bounded support introduced in [1].