arXiv subjects
Publications and source records attributed to Horst Martini.
We study the generalized analogues of conics for normed planes by using the following natural approach: It is well known that there are different metrical definitions of conics in the Euclidean plane. We investigate how these definitions extend to normed planes, and we show that in this more general framework these different definitions yield, in almost all cases, different classes of curves.
We give a tight upper bound on the polygonal diameter of the interior, resp. exterior, of a simple $n$-gon, $n \ge 3$, in the plane as a function of $n$, and describe an $n$-gon $(n \ge 3)$ for which both upper bounds (for the interior and the exterior) are attained \emph{simultaneously}.
Let V be a finite set of points in Euclidean d-space (d >= 2). The intersection of all unit balls B(v,1) centered at v, where v ranges over V, henceforth denoted by B(V) is the ball polytope associated with V. Note that B(V) is non-empty iff the circumradius of V is <= 1. After some preparatory discussion on spherical convexity and spindle convexity, the paper focuses on two central themes. [a] Define the boundary complex of B(V) (assuming it is non-empty, of course), i.e., define its vertices, edges and facets in dimension 3 (in dimension 2 this complex is just a circuit), and investigate its basic properties. [b] Apply results of this investigation to characterize finite sets of diameter 1 in (Euclidean) 3-space for which the diameter is attained a maximal number of times as a segment (of length 1) with both endpoints in V. A basic result for such a characterization goes back to Grunbaum, Heppes and Straszewicz, who proved independently that the diameter of V is attained at most 2|V|-2 times, thus affirming a conjecture of Vazsonyi from circa 1935. Call V extremal if its diameter is attained this maximal number (2|V|-2) of times. We extend the aforementioned basic result by showing that V is extremal iff V coincides with the set of vertices of its ball polytope B(V) and show that in this case the boundary complex of B(V) is self-dual in some strong sense. For the sake of priority we mention that, in the present form (except for a few changes in the footnotes), the paper was submitted to a journal already in February 1, 2008.
We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have often been rediscovered as lemmas to other results. In Part I we cover the following topics: The triangle inequality and consequences such as the monotonicity lemma, geometric characterizations of strict convexity, normality (Birkhoff orthogonality), conjugate diameters and Radon curves, equilateral triangles and the affine regular hexagon construction, equilateral sets, circles: intersection, circumscribed, characterizations, circumference and area, inscribed equilateral polygons.
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.
We give a complete proof that in any finite-dimensional normed linear space a finite set of points has a minimal spanning tree in which the maximum degree is bounded above by the strict Hadwiger number of the unit ball, i.e., the largest number of unit vectors such that the distance between any two is larger than 1.
A simplex is said to be orthocentric if its altitudes intersect in a common point, called its orthocenter. In this paper it is proved that if any two of the traditional centers of an orthocentric simplex (in any dimension) coincide, then the simplex is regular. Along the way orthocentric simplices in which all facets have the same circumradius are characterized, and the possible barycentric coordinates of the orthocenter are described precisely. In particular these barycentric coordinates are used to parametrize the shapes of orthocentric simplices. The substantial, but widespread, literature on orthocentric simplices is briefly surveyed in order to place the new results in their proper context, and some of the previously known results are given new proofs from the present perspective.
We investigate the geometric properties of simplices in Euclidean d-dimensional space for which two or more of the analogues of the classical triangle centers (including the centroid, circumcenter, incenter, orthocenter or Monge point, and the Fermat-Torricelli point) coincide. We also investigate the geometric significance of the cevian line segments through a given center having the same length. We give a unified presentation, including known results for d=2 and d=3.
In this paper we consider two notions that have been discovered and rediscovered by geometers and analysts since 1917 up to the present day: Radon curves and antinorms. A Radon curve is a special kind of centrally symmetric closed convex curve in the plane. A Radon plane is a normed plane obtained by using a Radon curve as (the boundary of) the unit ball. Many known results in Euclidean geometry also hold for Radon planes, for example the triangle and parallelogram area formulas, certain theorems on angular bisectors, the area formula of a polygon circumscribed about a circle, certain isoperimetric inequalities, and the non-expansive property of certain non-linear projections. These results may be further generalized to arbitrary normed planes if we formally change the statement of the result by referring in some places to the antinorm instead of the norm. The antinorm is a norm dual to the norm of an arbitrarily given normed plane, although it lives in the same plane as the original norm. It is the purpose of this mainly expository paper to give a list of results on antinorms that generalize results true for Radon norms, and in many cases characterize Radon norms among all norms in the plane. Many of the results are old, well-known, and have often been rediscovered. However, for most of the results we give streamlined proofs. Also, some of the characterizations of Radon curves given here seem not to have appeared previously in print.