  Stabilizer-free $P_k$ virtual elements are constructed on polygonal and
polyhedral meshes. Here the interpolating space is the space of continuous
$P_k$ polynomials on a triangular-subdivision of each polygon, or a
tetrahedral-subdivision of each polyhedron. With such an accurate and proper
interpolation, the stabilizer of the virtual elements is eliminated while the
system is kept positive-definite. We show that the stabilizer-free virtual
elements converge at the optimal order in 2D and 3D. Numerical examples are
computed, validating the theory.
