Inequalities for sections and projections of log-concave functions

Abstract: We provide extensions of geometric inequalities about sections and projections of convex bodies to the setting of integrable log-concave functions. Namely, we consider suitable generalizations of the affine and dual affine quermassintegrals of a log-concave function $f$ and obtain upper and lower estimates for them in terms of the integral $|f|_1$ of $f$, we give estimates for sections and projections of log-concave functions in the spirit of the lower dimensional Busemann-Petty and Shephard problem, and we extend to log-concave functions the affirmative answer to a variant of the Busemann-Petty and Shephard problems, proposed by V. Milman. The main goal of this article is to show that the assumption of log-concavity leads to inequalities in which the constants are of the same order as that of the constants in the original corresponding geometric inequalities.