Inequalities for Sections and Projections of Log-Concave Functions

Abstract We extend several geometric inequalities for sections and projections of convex bodies to the setting of integrable log-concave functions. In particular, we introduce suitable generalizations of the affine and dual affine quermassintegrals of a log-concave function f and obtain upper and lower bounds for them in terms of the integral $$\Vert f\Vert _1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>‖</mml:mo> <mml:mi>f</mml:mi> <mml:mo>‖</mml:mo> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:math> . We also establish estimates for sections and projections of log-concave functions in the spirit of the lower dimensional Busemann-Petty and Shephard problems and extend to the log-concave setting the affirmative answer to a variant of these problems, proposed by V. Milman. The main objective of this work is to show that the assumption of log-concavity leads to inequalities whose constants are of the same order as those in the corresponding geometric inequalities.