  In this paper we address two different related problems. We first study the
problem of finding a simple shortest path in a $d$-dimensional real space
subdivided in several polyhedra endowed with different $\ell_p$-norms. This
problem is a variant of the weighted region problem, a classical path problem
in computational geometry introduced in Mitchell and Papadimitriou (JACM
38(1):18-73, 1991). As done in the literature for other geodesic path problems,
we relate its local optimality condition with Snell's law and provide an
extension of this law in our framework space. We propose a solution scheme
based on the representation of the problem as a mixed-integer second order cone
problem (MISOCP) using the $\ell_p$-norm modeling procedure given in Blanco et
al. (Comput Optim Appl 58(3):563-595, 2014). We derive two different MISOCPs
formulations, theoretically compare the lower bounds provided by their
continuous relaxations, and propose a preprocessing scheme to improve their
performance. The usefulness of this approach is validated through computational
experiments. The formulations provided are flexible since some extensions of
the problem can be handled by transforming the input data in a simple way. The
second problem that we consider is the Weber problem that results in this
subdivision of $\ell_p$-normed polyhedra. To solve it, we adapt the solution
scheme that we developed for the shortest path problem and validate our
methodology with extensive computational experiments.
