  We extend to a functional setting the concept of quermassintegrals,
well-known within the Minkowski theory of convex bodies. We work in the class
of quasi-concave functions defined on the Euclidean space, and with the
hierarchy of their subclasses given by $\alpha$-concave functions. In this
setting, we investigate the most relevant features of functional
quermassintegrals, and we show they inherit the basic properties of their
classical geometric counterpart. As a first main result, we prove a
Steiner-type formula which holds true by choosing a suitable functional
equivalent of the unit ball. Then, we establish concavity inequalities for
quermassintegrals and for other general hyperbolic functionals, which
generalize the celebrated Pr\'ekopa-Leindler and Brascamp-Lieb inequalities.
Further issues that we transpose to this functional setting are:
integral-geometric formulae of Cauchy-Kubota type, valuation property and
isoperimetric/Uryshon-like inequalities.
