The dual Orlicz-Aleksandrov-Fenchel inequality

Aleksandrov-Fenchel inequality for coconvex bodies

We prove a version of the Aleksandrov-Fenchel inequality for mixed volumes of coconvex bodies. This version is motivated by an inequality from commutative algebra relating intersection multiplicities of ideals.

On mixed Hodge-Riemann relations for translation-invariant valuations and Aleksandrov-Fenchel inequalities

A version of the Hodge-Riemann relations for valuations was recently conjectured and proved in several special cases by the first-named author. The Lefschetz operator considered there arises as either the product or the convolution with the mixed volume of several Euclidean balls. Here we prove that in (co-)degree one the Hodge-Riemann relations persist if the balls are replaced by several different (centrally symmetric) convex bodies with smooth boundary with positive Gauss curvature. While these mixed Hodge-Riemann relations for the convolution directly imply the Aleksandrov-Fenchel inequali…

投影函数的拉东变换与亚历山德罗夫-芬切尔不等式

Radon Transforms of Projection Functions and Aleksandrov-Fenchel Inequalities

本文针对平均截面体的对数凹性问题，建立新的亚历山德罗夫-芬切尔不等式，证明平均截面体的所有内在体积均为对数凹的。

The mixed affine quermassintegrals

In this paper, we introduce first the mixed affine quermassintegrals. The Aleksandrov-Fenchel inequality for the mixed affine quermassintegrals is established. As an application, the Minkowski, Brunn-Minkowski inequalities for the mixed affine quermassintegrals are also derived.

Dual Orlicz-Brunn-Minkowski theory: dual Orlicz $L_ϕ$ affine and geominimal surface areas

This paper aims to develop basic theory for the dual Orlicz $L_ϕ$ affine and geominimal surface areas for star bodies, which belong to the recent dual Orlicz-Brunn-Minkowski theory for star bodies. Basic properties for these new affine invariants will be provided. Moreover, related Orlicz affine isoperimetric inequality, cyclic inequality, Santaló style inequality and Alexander-Fenchel type inequality are established.

Inequalities for dual quermassintegrals of mixed intersection bodies

In this paper, we first introduce a new concept of {\it dual quermassintegral sum function} of two star bodies and establish Minkowski's type inequality for dual quermassintegral sum of mixed intersection bodies, which is a general form of the Minkowski inequality for mixed intersection bodies. Then, we give the Aleksandrov--Fenchel inequality and the Brunn--Minkowski inequality for mixed intersection bodies and some related results. Our results present, for intersection bodies, all dual inequalities for Lutwak's mixed prosection bodies inequalities.