Title: Inequalities for sections and projections of log-concave functions Show Chinese title

Title: 对数凹函数的截面和投影的不等式

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. Show Chinese abstract

Abstract: 我们提供了关于凸体截面和投影的几何不等式的扩展，适用于可积的对数凹函数。 具体来说，我们考虑了对数凹函数$f$的仿射和对偶仿射广义体积积分的适当推广，并根据$f$的积分$\|f\|_1$给出了它们的上下界估计，我们在低维 Busemann-Petty 和 Shephard 问题的精神下给出了对数凹函数的截面和投影的估计，并将 V. Milman 提出的 Busemann-Petty 和 Shephard 问题的一个变种的肯定答案扩展到对数凹函数。 本文的主要目标是证明对数凹性的假设导致不等式，其中常数的阶与原始相应几何不等式中的常数的阶相同。