  We show that the entanglement entropy and alpha entropies corresponding to
spatial polygonal sets in $(2+1)$ dimensions contain a term which scales
logarithmically with the cutoff. Its coefficient is a universal quantity
consisting in a sum of contributions from the individual vertices. For a free
scalar field this contribution is given by the trace anomaly in a three
dimensional space with conical singularities located on the boundary of a plane
angular sector. We find its analytic expression as a function of the angle.
This is given in terms of the solution of a set of non linear ordinary
differential equations. For general free fields, we also find the small-angle
limit of the logarithmic coefficient, which is related to the two dimensional
entropic c-functions. The calculation involves a reduction to a two dimensional
problem, and as a byproduct, we obtain the trace of the Green function for a
massive scalar field in a sphere where boundary conditions are specified on a
segment of a great circle. This also gives the exact expression for the
entropies for a scalar field in a two dimensional de Sitter space.
