Abstract Representations of equiaffine surface area, due to Leichtweiß resp. Schütt & Werner,
are generalized top-affine surface area measures. We provide a direct proof which shows …

The Orlicz-Brunn-Minkowski theory, introduced by Lutwak, Yang, and Zhang, is a new
extension of the classical Brunn-Minkowski theory. It represents a generalization of the $L_p$-…

Random polytopes arise naturally as convex hulls of random points selected according to a
given distribution. In a dual way, they can be derived as intersections of random halfspaces. …

Convexity is an elementary property of sets and functions. A subset A of an affine space is
convex if it contains all the segments joining any two points of A. In other words, A is convex if …

Two new approaches are presented to establish the existence of polytopal solutions to the
discrete-data Lp Minkowski problem for all p> 1. As observed by Schneider [23], the Brunn-…

Predicting physical properties of materials with spatially complex structures is one of the
most challenging problems in material science. One key to a better understanding of such …

This paper describes the theoretical foundation of and explicit algorithms for a novel
approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued …

We prove a complete set of integral geometric formulas of Crofton type (involving integrations
over affine Grassmannians) for the Minkowski tensors of convex bodies. Minkowski tensors …

A first step towards a dual Orlicz–Brunn–Minkowski theory for star sets was taken by Zhu,
Zhou, and Xue [44], [45]. In this essentially independent work we provide a more general …

The general volume of a star body, a notion that includes the usual volume, the qth dual
volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new …