In this paper we consider two notions that have been discovered and rediscovered by geometers and analysts since 1917 up to the present day. The first is that of a Radon curve, introduced by Radon in 1917 [50]. It is a special kind of centrally symmetric closed convex curve in the plane. Any centrally symmetric closed convex curve in the plane defines a norm turning the plane into a two-dimensi...

Soil structure is developed across a wide range of scales including pores of different origin and herewith different size and shape. At larger scales soil structure is expressed in a heterogeneous distribution of local bulk densities. X-ray tomography is a suitable tool to measure and visualize this hierarchical structure at various scales and resolutions. However, because of the wide range of ...

We define complex Minkowski superspace in 4 dimensions as the big cell inside a complex flag supermanifold. The complex conformal supergroup acts naturally on this super flag, allowing us to interpret it as the conformal compactification of complex Minkowski superspace. We then consider real Minkowski superspace as a suitable real form of the complex version. Our methods are group theoretic, ba...

Analogs of the classical inequalities from the Brunn Minkowski Theory for rotation intertwining additive maps of convex bodies are developed. We also prove analogs of inequalities from the dual Brunn Minkowski Theory for intertwining additive maps of star bodies. These inequalities provide generalizations of results for projection and intersection bodies. As a corollary we obtain a new Brunn Mi...

In this paper we construct a new family of flat Minkowski planes of group dimension 3. These planes share the positive half with the classical real Minkowski plane and admit simple groups of automorphisms isomorphic to PSL2ðRÞ acting diagonally on the torus. We further determine the full automorphism groups and the Klein–Kroll types of these flat Minkowski planes. 2000 Mathematics Subject Class...

Article history: Available online 17 July 2008

The Brunn-Minkowski inequality theory plays an important role in a number of mathematical disciplines such as measure theory, crystallography, optimal control theory, functional analysis, and geometric convexity. It has many useful applications in combinatorics, stochastic geometry, and mathematical economics. In recent years, several authors including Ball [1, 2, 3], Bourgain and Lindenstrauss...

A volume-normalized formulation of the Lp-Minkowski problem is presented. This formulation has the advantage that a solution is possible for all p ≥ 1, including the degenerate case where the index p is equal to the dimension of the ambient space. A new approach to the Lp-Minkowski problem is presented, which solves the volume-normalized formulation for even data and all p ≥ 1. The Minkowski pr...

Centroid and difference bodies define SL(n) equivariant operators on convex bodies and these operators are valuations with respect to Minkowski addition. We derive a classification of SL(n) equivariant Minkowski valuations and give a characterization of these operators. We also derive a classification of SL(n) contravariant Minkowski valuations and of Lp-Minkowski valuations. 2000 AMS subject c...

We introduce and study Minkowski games. In these games, two players take turns to choose positions in R based on some rules. Variants include boundedness games, where one player wants to keep the positions bounded (while the other wants to escape to infinity), and safety games, where one player wants to stay within a given set (while the other wants to leave it). We provide some general charact...