Gate points in continuous location between regions with different ℓp norms

In this paper, a set of notable points is given from the ℓp-distance between a point P and a straight line r  . The problem of the geodesic path between two points of R2R2 separated by a straight line r with different norms in each half-plane is then studied from the notable points of these points. This geodesic path provides a point in r, called the “Gate point”, which is studied by varying ℓp- and ℓq-distances, and the slope m of the straight line r. The analytical expression of the Gate point is given in some particular cases.

► A set of notable points is given from the ℓp-distance between a point P and a line r.
► “Gate point”, is found by varying ℓp and ℓq-distances.
► The analytical expression of the Gate point is given in some particular cases.
► A more general result (Theorem 2.10) of Lemma 1 in [3] and [18] is obtained for ℓp and ℓq.
► The bound attained is also tighter than in [3] and [18].