In this paper we prove some results on Reuleaux triangles in (Minkowski or) normed planes. For example, we reprove Wernicke’s result (see [21]) that the unit disc and Reuleaux triangles in a normed plane are homothets if and only if the unit circle is either an affine regular hexagon or a parallelogram. Also we show that the ratio of the area of the unit ball of a Minkowski plane to that of a R...

Minkowski sums cover a wide range of applications in many different fields like algebra, morphing, robotics, mechanical CAD/CAM systems ... This paper deals with sums of polytopes in a n dimensional space provided that both H-representation and V-representation are available i.e. the polytopes are described by both their half-spaces and vertices. The first method uses the polytope normal fans a...

We obtain a new characterization of the diametrically complete sets in Minkowski spaces, by modifying two well-known characteristic properties of bodies of constant width. We also get sharp inequalities for the circumradius and inradius of a diametrically complete set of given diameter. Strengthening former work of D. Yost, we show that in a generic Minkowski space of dimension at least three t...

Non-extreme walls (bubbles with two insides) and ultra-extreme walls (bubbles of false vacuum decay) are discussed. Their respective energy densities are higher and lower than that of the corresponding extreme (supersymmetric), planar domain wall. These singularity free space-times exhibit non-trivial causal structure analogous to certain non-extreme black holes. We focus on anti-de Sitter– Min...

The repertoire of set operators available in Constructive Solid Geometry (CSG) may be extended by the Minkowski addition and subtraction operators. In this way the problem of visualizing 3D solids deened as a Minkowski addition or subtraction of elementary objects can be solved by using CSG in combination with ray tracing. The method consists in transforming a Minkowski term with multiple opera...

The objective of this paper is to present two types of results on Minkowski sums of convex polytopes. The first is about a special class of polytopes we call perfectly centered and the combinatorial properties of the Minkowski sum with their own dual. In particular, we have a characterization of face lattice of the sum in terms of the face lattice of a given perfectly centered polytope. Exact f...

In this paper we introduce the Minkowski weighted partition around medoids algorithm (MW-PAM). This extends the popular partition around medoids algorithm (PAM) by automatically assigning K weights to each feature in a dataset, where K is the number of clusters. Our approach utilizes the within-cluster variance of features to calculate the weights and uses the Minkowski metric. We show through ...

The Hasse-Minkowski Theorem provides a characterization of the rational quadratic forms. What follows is a proof of the Hasse-Minkowski Theorem paraphrased from the book, Number Theory by Z.I. Borevich and I.R. Shafarevich [1]. Throughout this paper, some familiarity with the p-adic numbers and the Hilbert symbol is assumed and some basic facts about quadratic forms are stated without proof. Al...

This is a supplement to our previous research note [3]. In [3], we gave a characterization of biharmonic curves in Minkowski 3-space. More precisely, we pointed out that every biharmonic curves with nonnull principal normal in Minkowski 3-space is a helix, whose curvature κ and torsion τ satisfy κ2 = τ2. In the classification of biharmonic curves in Minkowski 3-space due to Chen and Ishikawa [1...

Abstract This paper aims to consider the dual Brunn–Minkowski inequality for log-volume of star bodies, and equivalent Minkowski mixed log-volume.