  In the present contribution we develop a sharper error analysis for the
Virtual Element Method, applied to a model elliptic problem, that separates the
element boundary and element interior contributions to the error. As a
consequence we are able to propose a variant of the scheme that allows to take
advantage of polygons with many edges (such as those composing Voronoi meshes
or generated by agglomeration procedures) in order to yield a more accurate
discrete solution. The theoretical results are supported by numerical
experiments.
