  The contributions to the heat kernel coefficients generated by the corners of
the boundary are studied. For this purpose the internal and external sectors of
a wedge and a cone are considered. These sectors are obtained by introducing,
inside the wedge, a cylindrical boundary. Transition to a cone is accomplished
by identification of the wedge sides. The basic result of the paper is the
calculation of the individual contributions to the heat kernel coefficients
generated by the boundary singularities. In the course of this analysis certain
patterns, that are followed by these contributions, are revealed. The
implications of the obtained results in calculations of the vacuum energy for
regions with nonsmooth boundary are discussed. The rules for obtaining all the
heat kernel coefficients for the minus Laplace operator defined on a polygon or
in its cylindrical generalization are formulated.
