When performing geometrical analysis of shapes, the thickness of the shape at a point is a fundamental quantity both for theoretical and practical purposes. In this paper, we consider a thickness definition which associates a value with each point of the shape (i.e. both on its boundary and interior). If we now consider a discrete shape characterized by a sampling of a continuous shape boundary, such thickness function on interior points induces a combinatorial structure which can be used to efficiently compute some relevant quantities for shape description. The aim of the paper is thus to introduce and analyse the combinatorial structure, and to provide some convergence results when the sampling gets denser since the thickness function is deeply linked to the medial axis of the shape.