  Building on an idea laid out by Martelli--Sparks--Yau, we use the
Duistermaat-Heckman localization formula and an extension of it to give
rational and explicit expressions of the volume, the total transversal scalar
curvature and the Einstein--Hilbert functional, seen as functionals on the
Sasaki cone (Reeb cone). Studying the leading terms we prove they are all
proper. Among consequences we get that the Einstein-Hilbert functional attains
its minimal value and each Sasaki cone possess at least one Reeb vector field
with vanishing transverse Futaki invariant.
