Related papers: On generalized Minkowski arrangements
In this paper, we generalize the Minkowski distance by defining a new distance function in n-dimensional space, and we show that this function determines also a metric family as the Minkowski distance. Then, we consider three special cases…
An important theorem of Banaszczyk (Random Structures & Algorithms `98) states that for any sequence of vectors of $\ell_2$ norm at most $1/5$ and any convex body $K$ of Gaussian measure $1/2$ in $\mathbb{R}^n$, there exists a signed…
Suppose one has a collection of disks of various sizes with disjoint interiors, a packing in the plane, and suppose the ratio of the smallest radius divided by the largest radius lies between $1$ and $q$. In his 1964 book Regular Figures…
In "Weighted Brunn-Minkowski Theory I", the prequel to this work, we discussed how recent developments on concavity of measures have laid the foundations of a nascent weighted Brunn-Minkowski theory. In particular, we defined the mixed…
The affine quermassintegrals associated to a convex body in $\mathbb{R}^n$ are affine-invariant analogues of the classical intrinsic volumes from the Brunn-Minkowski theory, and thus constitute a central pillar of affine convex geometry.…
In this paper we investigate the problem of finding the minimum edge density in families of convex, normal mosaics with unit volume cells in $n$-dimensional Euclidean space. In the first part of the paper we solve this problem for mosaics…
Minkowski's second theorem on successive minima gives an upper bound on the volume of a convex body in terms of its successive minima. We study the problem to generalize Minkowski's bound by replacing the volume by the lattice point…
Analogues of the classical inequalities from the Brunn-Minkowski theory for rotation intertwining additive maps of convex bodies are developed. Analogues are also proved of inequalities from the dual Brunn-Minkowski theory for intertwining…
Let $f:\mathbb{P}^1\to\mathbb{P}^1$ be a rational map of degree $d\geq2$ defined over a number field $K$ and let $\alpha\in\mathbb{P}^1(K)$. We consider the lower and upper Minkowski dimensions of the arboreal Galois group $G_{f,\alpha}$…
Every polyhedron can be decomposed into a Minkowski sum (or vector sum) of a bounded polyhedron and a polyhedral cone. This paper establishes similar statements for some classes of discrete sets in discrete convex analysis, such as…
In this survey we present a generalization of the notion of metric space and some applications to discrete structures as graphs, ordered sets and transition systems. Results in that direction started in the middle eighties based on the…
The local Minkowski tensors are valuations on the space of convex bodies in Euclidean space with values in a space of tensor measures. They generalize at the same time the intrinsic volumes, the curvature measures and the isometry covariant…
We investigate Minkowski additive, continuous, and translation invariant operators $\Phi:\mathcal{K}^n\to\mathcal{K}^n$ defined on the family of convex bodies such that the volume of the image $\Phi(K)$ is bounded from above and below by…
The Brunn-Minkowski theory relies heavily on the notion of mixed volumes. Despite its particular importance, even explicit representations for the mixed volumes of two convex bodies in Euclidean space are available only in special cases.…
We establish the following max-plus analogue of Minkowski's theorem. Any point of a compact max-plus convex subset of $(R\cup\{-\infty\})^n$ can be written as the max-plus convex combination of at most $n+1$ of the extreme points of this…
It is a classical fact, that given an arbitrary n-dimensional convex body, there exists an appropriate sequence of Minkowski symmetrizations (or Steiner symmetrizations), that converges in Hausdorff metric to a Euclidean ball. Here we…
For a given partially ordered set (poset) and a given family of mappings of the poset into itself, we study the problem of the description of joint fixed points of this family. Well-known Tarski's theorem gives the structure of the set of…
We analyse the geometry of generic Minkowski $\mathcal{N}=1$, $D=4$ flux compactifications in string theory, the default backgrounds for string model building. In M-theory they are the natural string theoretic extensions of $\mathrm{G}_2$…
The classical Minkowski inequality in the Euclidean space provides a lower bound on the total mean curvature of a hypersurface in terms of the surface area, which is optimal on round spheres. In this paper we employ a locally constrained…
In this paper we show that every sufficiently large family of convex bodies in the plane has a large subfamily in convex position provided that the number of common tangents of each pair of bodies is bounded and every subfamily of size five…