Minkowski bisectors,Minkowski cells and lattice coverings

We prove Barth-type connectedness results for low-codimension smooth subvarieties with good numerical properties inside certain “easy” ambient spaces (such as homogeneous varieties, or spherical varieties). The argument employs some basics from the theory of cones of cycle classes, in particular the notion of bigness of a cycle class.     

Hermann Minkowski

Ohne ZusammenfassungGedächtnisrede, gehalten in der öffentlichen Sitzung der Kgl. Gesellschaft der Wissenschaften zu Göttingen am 1. Mai 1909.     

Minkowski valuations

Centroid and difference bodies define equivariant operators on convex bodies and these operators are valuations with respect to Minkowski addition. We derive a classification of equivariant Minkowski valuations and give a characterization of these operators. We also derive a classification of contravariant Minkowski valuations and of -Minkowski valuations.

     

Representation of projectors involving Minkowski inverse in Minkowski space

A certain class of results about the different representations of Oblique projectors is present in the literature. These results represent Oblique projectors as the functions of orthogonal projectors with given onto and along spaces. But these results are valid under the restriction that the functions of orthogonal projectors involved are invertible. In this paper we extend and generalize these results. The extension lies in making a transition from Euclidean space to Minkowski space M and the generalization is obtain by voiding the invertibility condition and use of the Minkowski inverse. Furthermore, the nobility lies in utilizing the m-projectors instead of the regular orthogonal projectors.     

Minkowski content and local Minkowski content for a class of self-conformal sets

We investigate (local) Minkowski measurability of ${\mathcal {C}^{1+\alpha}}$ images of self-similar sets. We show that (local) Minkowski measurability of a self-similar set K implies (local) Minkowski measurability of its image F and provide an explicit formula for the (local) Minkowski content of F in this case. A counterexample is presented which shows that the converse is not necessarily true. That is, F can be Minkowski measurable although K is not. However, we obtain that an average version of the (local) Minkowski content of both K and F always exists and also provide an explicit formula for the relation between the (local) average Minkowski contents of K and F.     

Finite Minkowski planes

In this paper we give second characterizations of a certain class of finite Minkowski planes.     

Projective Minkowski set

The Minkowski set or the central symmetry set (CSS) of a smooth curve Γ on the affine plane is the envelope of chords connecting pairs of points such that the tangents to Γ at them are parallel. Singularities of CSS are of interest, in particular, for applications (for example, in computer graphics). A generalization of the Minkowski set is considered in the paper, namely, the projective Minkowski set with respect to a line on the plane; in the case of general position, we describe its singularities and the bifurcation set of lines corresponding to lines defining the projective Minkowski set having singularities being more degenerate than those of the Minkowski set for a generic line.     

Intrinsic volumes and successive radii

Motivated by a problem of Teissier to bound the intrinsic volumes of a convex body in terms of the inradius and the circumradius of the body, we give upper and lower bounds for the intrinsic volumes of a convex body in terms of the elementary symmetric functions of the so-called successive inner and outer radii. These results improve on former bounds and, in particular, they also provide bounds for the elementary symmetric functions of the roots of Steiner polynomials in terms of the elementary symmetric functions of these radii.     

Minkowski operations and vector spaces

Mathematical morphology started as a set of tools for analysing images by the use of transformations based on set-theoretical operations which are the Minkowski sum and subtraction. It was first developed for the analysis of binary images. Its extension to grey-level images was a later development with the extension of the Minkowski operations to real-valued functions in terms of sup-convolution and inf-convolution. The purpose of this paper is to define a type of convolution between set-valued maps, to study its properties, and to establish some associated differential relations. This set-convolution map allows us to extend the Minkowski sum and substraction to multivalued functions and to functions with vectorial values.     

Hilbert metrics and Minkowski norms

It is shown that the Hilbert geometry (D, h_D) associated to a bounded convex domain is isometric to a normed vector space if and only if D is an open n-simplex. One further result on the asymptotic geometry of Hilbert’s metric is obtained with corollaries for the behavior of geodesics. Finally we prove that every geodesic ray in a Hilbert geometry converges to a point of the boundary.     

Minkowski parallel-translation and their coordinatization

In a Minkowski plane with parallelisms ₊ and _– one postulates triple transitivity of ₊ -preserving bijections of points onto _– points such that circles map on circles. This postulate corresponds almost exactly to Axiom G of [2]. A coordinatization may then be accomplished by introduction of a quaternary operation, after a coordinate frame consisting of three distinct concircular points has been fixed. All circles except those in one derived affine plane are points (x,y) with y=[x,a,b,c]. This derived plane then turns out to be a nearfield-plane. The quaternary operation satisfies [x,a,b,c]=[x–a,0,0,1]c+b. In a symmetric Minkowski plane, which, in view of [1], is coordinatized by a commutative field, y=[x,a,b,c] becomes (x–a)(y–b)=c, yielding the well known hyperbola model.Dedicated to Werner Burau on his 70th birthdayPresented to the Conference on Geometry, University of Haifa, March 10, 1975.     

Two-transitive Minkowski planes

In this paper we determine all finite Minkowski planes with an automorphism group which satisfies the following transitivity property: any ordered pair of nonparallel points can be mapped onto any other ordered pair of nonparallel points.