Extension of the total mass of log-concave functions and related inequalities

The aim of this paper is to deal with the log-concave functions in $\mathbb R^n$, endowed with a suitable algebraic structure corresponds to the structure of convex bodies in $\mathbb R^n$, when restricted to the subclass of characteristic functions. In this paper, the functional Quermassintegrals of log-concave functions in $\mathbb R^n$ are discussed, we express the functional mixed Quermassintegrals as the integral of the support function of $f$ on some measures. We obtain a functional counterpart of the mixed Quermassintegrals inequality for convex bodies. Moreover, as a special case a weak log Quermassintegral inequality is obtained.