  We consider the problem of computing the Lebesgue volume of compact basic
semi-algebraic sets. In full generality, it can be approximated as closely as
desired by a converging hierarchy of upper bounds obtained by applying the
Moment-SOS (sums of squares) methodology to a certain infinite-dimensional
linear program (LP). At each step one solves a semidefinite relaxation of the
LP which involves pseudo-moments up to a certain degree. Its dual computes a
polynomial of same degree which approximates from above the discontinuous
indicator function of the set, hence with a typical Gibbs phenomenon which
results in a slow convergence of the associated numerical scheme. Drastic
improvements have been observed by introducing in the initial LP additional
linear moment constraints obtained from a certain application of Stokes'
theorem for integration on the set. However and so far there was no rationale
to explain this behavior. We provide a refined version of this extended LP
formulation. When the set is the smooth super-level set of a single polynomial,
we show that the dual of this refined LP has an optimal solution which is a
continuous function.Therefore in this dual one now approximates a continuous
function by a polynomial, hence with no Gibbs phenomenon, which explains and
improves the already observed drastic acceleration of the convergence of the
hierarchy. Interestingly, the technique of proof involves recent results on
Poisson's partial differential equation (PDE).
