Separating nn-point sets from quasi-finite ones via polyhedral surfaces

Let XX be an infinite set in RdRd that has no accumulation point. We prove that the following statement holds for each dd-dimensional polyhedron ΠΠ, i.e., for each bounded part of RdRd generated by a closed polyhedral surface: for any positive integer nn, there is a polyhedron similar to ΠΠ that contains exactly nn points taken from XX.