P-curve: An improved solution for the single-facility location problem in two regions with $\ell_{p}$- and $\ell_{q}$-norms

The distance between two points is provided by the length of their test connecting path. However, there is no unique way to attain this path in the space $R^2$, which is split by a straight line $L$ into two regions, $Ω_p$ and $Ω_q$, with $\ell_{p}$- and $\ell_q$-norms, respectively, where $1<p\leq