Abstract

In this paper we investigate geometric and analytical properties of p-convexity for sets in Rn, with 0 < p < 1. We introduce a generalized notion of distance associated with pconvex structures and analyze its fundamental properties, including lower semicontinuity, coercivity, existence and uniqueness of minimizers, and Lipschitz-type continuity with respect to the base point. This generalized distance recovers the classical Euclidean distance in the limiting case p = 1 and provides a natural extension to nonconvex configurations governed by a p-convex geometry. As an illustrative example, we consider a scalar obstacle-avoiding version of the generalized p-distance, where admissible curves are constrained to remain outside a prescribed closed set.