The Hadwiger theorem on convex functions, IV: The Klain approach

The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge-Ampère measures

A complete family of functional Steiner formulas is established. As applications, an explicit representation of functional intrinsic volumes using special mixed Monge-Ampère measures and a new version of the Hadwiger theorem on convex functions are obtained.

凸函数的 Hadwiger 定理，II：Cauchy-Kubota 公式

The Hadwiger theorem on convex functions, II: Cauchy-Kubota formulas

研究通过新的泛函 Cauchy-Kubota 公式建立凸函数 Hadwiger 定理，给出泛函内蕴体积的显式表示，并揭示其与经典内蕴体积的联系及非负赋值分类。

关于凸函数上向量值估值的Klain-Schneider定理

A Klain-Schneider Theorem for Vector-Valued Valuations on Convex Functions

本文建立了一个关于凸函数上向量值估值的功能类比Klain-Schneider定理，对连续、平移协变且简单的估值进行分类。在额外的旋转等变假设下，特征化了动量向量的分析对应物以及一个新的上皮平移不变估值。前者作为一族函数内在矩的最高次算子出现，这些矩通过平移与函数内在体积相关联。后者代表了一类Minkowski向量的最高次算子，这些向量在本文中引入，它们在凸体上没有经典对应物，因为它们由于Minkowski关系而消失。此外，还获得了关于极值次数的同质估值的进一步分类结果。

凸函数上的 Hadwiger 定理，I

The Hadwiger theorem on convex functions, I

本文对 ${\mathbb R}^n$ 上超强制凸函数空间中所有连续、上图平移与旋转不变的对合进行了完整分类，通过引入奇异 Hessian 对合，将其定义为经典内蕴体积的函数版本。

凸函数上的向量值 Hadwiger 定理

The Vectorial Hadwiger Theorem on Convex Functions

本文完成了凸函数上满足对偶上边缘平移不变与旋转等变的赋值的分类，刻画了函数性 Minkowski 向量，并给出奇异密度及混合 Monge-Ampère 测度等表示，同时推广到超强制情形。

Towards Hadwiger's conjecture via Bourgain Slicing

In 1957, Hadwiger conjectured that every convex body in $\mathbb{R}^d$ can be covered by $2^d$ translates of its interior. For over 60 years, the best known bound was of the form $O(4^d \sqrt{d} \log d)$, but this was recently improved by a factor of $e^{Ω(\sqrt{d})}$ by Huang, Slomka, Tkocz and Vritsiou. In this note we take another step towards Hadwiger's conjecture by deducing an almost-exponential improvement from the recent breakthrough work of Chen, Klartag and Lehec on Bourgain's slicing problem. More precisely, we prove that, for any convex body $K \subset \mathbb{R}^d$, $$\exp\bigg( -…