Some　error　analyses　on　virtual　element　methods　(VEMs)　including　inverse　inequalities,　norm　equivalence,　and　interpolation　error　estimates　are　developed　for　polygonal　meshes,　each　element　of　which　admits　a　virtual　quasi-uniform　triangulation.　This　sub-mesh　regularity　covers　the　usual　ones　used　for　theoretical　analysis　of　VEMs,　and　the　proofs　are　presented　by　means　of　standard　technical　tools　in　finite　element　methods.