Horst Martini

The geometry of Minkowski spaces - a survey. Part I

In this second part of a series of surveys on the geometry of finite dimensional Banach spaces (Minkowski spaces) we discuss results that refer to the following three topics: bodies of constant Minkowski width, generalized convexity notions that are important for Minkowski spaces, and bisectors as well as Voronoi diagrams in Minkowski spaces.

Excursions into combinatorial geometry

The book deals with the combinatorial geometry of convex bodies in finite-dimensional spaces. A general introduction to geometric convexity is followed by the investigation of d-convexity and H-convexity, and by various applications. Recent research is discussed, for example the three (generally unsolved) problems from the combinatorial geometry of convex bodies: the Szoekefalvi-Nagy problem, the Borsuk problem, the Hadwiger covering problem. These and related questions are then applied to a new class of convex bodies which is a natural generalization of the class of zonoids: the class of belt bodies. Finally open research problems are discussed.

Geometric methods and optimization problems

I. Nonclassical Variational Calculus. II. Median Problems in Location Science. III. Minimum Convex Partitions of Polygonal Domains.

The Fermat-Torricelli problem in normed planes and spaces

We investigate the Fermat–Torricelli problem in d-dimensional real normed spaces or Minkowski spaces, mainly for d=2. Our approach is to study the Fermat–Torricelli locus in a geometric way. We present many new results, as well as give an exposition of known results that are scattered in various sources, with proofs for some of them. Together, these results can be considered to be a minitheory of the Fermat–Torricelli problem in Minkowski spaces and especially in Minkowski planes. This demonstrates that substantial results about locational problems valid for all norms can be found using a geometric approach.

Antinorms and Radon curves

Summary.A Radon curve can be used as the unit circle of a norm, with the corresponding normed plane called a Radon plane. An antinorm is a special case of the Minkowski content of a measurable set in a Minkowski space. There is a long list of known results in Euclidean geometry that also hold for Radon planes. These results may sometimes be further generalized to arbitrary normed planes if we formally change such a statement by referring in some places to the antinorm instead of the norm. We present a list of such results for antinorms. Although most of these results are well known, we give streamlined proofs, and show which of these results lead to characterizations of Radon norms.As new results we prove two characterizations of Radon curves, one in terms of bisectors and the other in terms of triangles circumscribed about circles. We also solve the Zenodorus problem for Minkowski planes, i.e., we characterize the polygons with n sides that have the largest area for a fixed perimeter in any given Minkowski plane.

Median hyperplanes in normed spaces — a survey

In this survey we deal with the location of hyperplanes in n-dimensional normed spaces, i.e., we present all known results and a unifying approach to the so-called median hyperplane problem in Minkowski spaces. We describe how to find a hyperplane H minimizing the weighted sum f(H) of distances to a given, finite set of demand points. In robust statistics and operations research such an optimal hyperplane is called a median hyperplane. After summarizing the known results for the Euclidean and rectangular situation, we show that for all distance measures d derived from norms one of the hyperplanes minimizing f(H) is the affine hull of n of the demand points and, moreover, that each median hyperplane is a halving one (in a sense defined below) with respect to the given point set. Also an independence of norm result for finding optimal hyperplanes with fixed slope will be given. Furthermore, we discuss how these geometric criteria can be used for algorithmical approaches to median hyperplanes, with an extra discussion for the case of polyhedral norms. And finally a characterization of all smooth norms by a sharpened incidence criterion for median hyperplanes is mentioned.

Minimal enclosing discs, circumcircles, and circumcenters in normed planes (Part I)

It is surprising that there are almost no results on the precise location of (all) minimal enclosing balls, circumballs, and circumcenters of simplices in finite-dimensional real Banach spaces. In this paper and a subsequent second part of it we give the starting point in this direction, also for computational investigations. More precisely, we present the first thorough study of these topics for triangles in arbitrary normed planes. In the present Part I we lay special emphasize on a complete description of possible locations of the circumcenters, and as a needed tool we give also a modernized classification of all possible shapes of the intersection that two homothetic norm circles can create. Based on this, we give in Part II the complete solution of the strongly related subject to find all minimal enclosing discs of triangles in arbitrary normed planes.

Antipodality properties of finite sets in Euclidean space

This is a survey of known results and still open problems on antipodal properties of finite sets in Euclidean space. The exposition follows historical lines and takes into consideration both metric and affine aspects.

On the Number of Maximal Regular Simplices Determined by n Points in Rd

A set V = {x 1,…, x n } of n distinct points in Euclidean d-space ℝ d determines 2 n distances ∥x j − x i ∥ (1 ≤ i < j ≤ n). Some of these distances may be equal. Many questions concerning the distribution of these distances have been asked (and, at least partially, answered). E.g., what is the smallest possible number of distinct distances, as a function of d and n? How often can a particular distance (say, one) occur and, in particular, how often can the largest (resp., the smallest) distance occur?

The Fermat-Torricelli point and isosceles tetrahedra

An analytical, unifying approach to geometric properties of the Fermat-Torricelli point of four affinely independent points yields several characterizations of isosceles tetrahedra and, in particular, a characterization or regular tetrahedra within the set of isosceles tetrahedra by means of the solid angle sum.