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In this paper we discuss theoretical foundations of developing general methods for volume-based approximation of three-dimensional solids. We construct an iterative method that can be used for approximation of regular subsets of R^d (d Î N) in particular R³. We will define solid meshes and investigate the connection between solid meshes, regular sets and polyhedra. First the general description of the method will be given. The main idea of our algorithm is a kind of space partitioning with increasing atomic s-algebra sequences. In every step one atom will be divided into two nonempty atoms. We define a volume based distance metric and we give sufficient conditions for the convergence and monotonicity of the method. We show a possible application, a polyhedral approximation (or approximate convex decomposition) of triangular meshes.

Mathematics Subject Classification (2010): 41A35, 41A63.

Keywords: Solid mesh, regular sets, space partitioning, orthogonal projection, best approximation.