On regular 4-coverings and their application for lattice coverings in normed planes

It is well known that the famous covering problem of Hadwiger is completely solved only in the planar case, i.e.: any planar convex body can be covered by four smaller homothetical copies of itself. Lassak derived the smallest possible ratio of four such homothets (having equal size), using the notion of regular 4-covering. We will continue these investigations, mainly (but not only) referring to centrally symmetric convex plates. This allows to interpret and derive our results in terms of Minkowski geometry (i.e., the geometry of finite dimensional real Banach spaces). As a tool we also use the notion of quasi-perfect and perfect parallelograms of normed planes, which do not differ in the Euclidean plane. Further on, we will use Minkowskian bisectors, different orthogonality types, and further notions from the geometry of normed planes, and we will construct lattice coverings of such planes and study related Voronoi regions and gray areas. Discussing relations to the known bundle theorem, we also extend Miquel’s six-circles theorem from the Euclidean plane to all strictly convex normed planes.     

Overdetermined elliptic problems in topological disks

We introduce a method, based on the Poincaré–Hopf index theorem, to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk. Applications to some well-known nonlinear elliptic PDEs are provided. Our result can be seen as the analogue of Hopf's uniqueness theorem for constant mean curvature spheres, but for the general analytic context of overdetermined elliptic problems.     

Local compactness in right bounded asymmetric normed spaces

We characterize the finite dimensional asymmetric normed spaces which are right bounded and the relation of this property with the natural compactness properties of the unit ball, such as compactness and strong compactness. In contrast with some results found in the existing literature, we show that not all right bounded asymmetric norms have compact closed balls. We also prove that there are finite dimensional asymmetric normed spaces that satisfy that the closed unit ball is compact, but not strongly compact, closing in this way an open question on the topology of finite dimensional asymmetric normed spaces. In the positive direction, we will prove that a finite dimensional asymmetric normed space is strongly locally compact if and only if it is right bounded.     

On a lower and upper bound for the curvature of ellipses with more than two foci

Let a set of points in the Euclidean plane be given. We are going to investigate the levels of the function measuring the sum of distances from the elements of the pointset which are called foci. Levels with only one focus are circles. In case of two different points as foci they are ellipses in the usual sense. If the set of the foci consists of more than two points then we have the so-called polyellipses. In this paper we investigate them from the viewpoint of differential geometry. We give a lower and upper bound for the curvature involving explicit constants. They depend on the number of the foci, the rate of the level and the global minimum of the function measuring the sum of the distances. The minimizer will be characterized by a theorem due to E. Weiszfeld together with a new proof. Explicit examples will also be given. As an application we present a new proof for a theorem due to P. Erd?s and I. Vincze. The result states that the approximation of a regular triangle by circumscribed polyellipses has an absolute error in the sense that there is no way to exceed it even if the number of the foci are arbitrary large.     

An extension of a theorem of Serrin to graphs in warped products

In this article we extend a well known theorem of J. Serrin about existence and uniqueness of graphs of constant mean curvature in Euclidean space to a broad class of Riemannian manifolds. Our result also generalizes several others proved recently and includes the new case of Euclidean “rotational” graphs with constant mean curvature.     

Random strict convexity and random uniform convexity in random normed modules

Based on the analysis of stratification structure on random normed modules, we first present random strict convexity and random uniform convexity in random normed modules. Then, we establish their respective relations to classical strict and uniform convexity: in the process some known important results concerning strict convexity and uniform convexity of Lebesgue-Bochner function spaces can be obtained as a special case of our results. Further, we also give their important applications to the theory of random conjugate spaces as well as best approximation. Finally, we conclude this paper with some remarks showing that the study of geometry of random normed modules will also motivate the further study of geometry of probabilistic normed spaces.     

Some mathematical problems in computer vision

Several interesting mathematical problems arising in computer vision are discussed. Computer vision deals with image understanding at various levels. At the low level, it addresses issues like segmentation, edge detection, planar shape recognition and analysis. Classical results on differential invariants associated to planar curves are relevant to planar object recognition under partial occlusion, and recent results concerning the evolution of closed planar shapes under curvature controlled diffusion have found applications in shape decomposition and analysis. At higher levels, computer vision problems deal with attempts to invert imaging projections and shading processes toward depth recovery, spatial shape recognition and motion analysis. In this context, the recovery of depth from shaded images of objects with smooth, diffuse surfaces require the solution of nonlinear partial differential equations. Here results on differential equations, as well as interesting results from low-dimensional topology and differential geometry are the necessary tools of the trade. We are still far from being able to equip our computers with brains capable to analyze and understand the images that can easily be acquired with camera-eyes; however the research effort in this area often calls for both classical and recent mathematical results.This work was supported in part by NSF grant DMS-8811084, Air Force Office of Scientific Research Grant AFOSR-90-0024, and the Army Research Office DAAL03-91-G-0019, and by the Technion Fund for Promotion of Research.     

A Mazur–Ulam theorem in non-Archimedean normed spaces

The classical Mazur–Ulam theorem which states that every surjective isometry between real normed spaces is affine is not valid for non-Archimedean normed spaces. In this paper, we establish a Mazur–Ulam theorem in the non-Archimedean strictly convex normed spaces.     

A Möbius characterization of submanifolds in real space forms with parallel mean curvature and constant scalar curvature

In this paper, we give a Möbius characterization of submanifolds in real space forms with parallel mean curvature vector fields and constant scalar curvatures, generalizing a theorem of H. Li and C.P. Wang in [LW1].Supported by NSF of Henan, P. R. China     

Constant Minkowskian width in terms of double normals

We extend the notion of a double normal of a convex body from smooth, strictly convex Minkowski planes to arbitrary two-dimensional real, normed, linear spaces in two different ways. Then, for both of these ways, we obtain the following characterization theorem: a convex body K in a Minkowski plane is of constant Minkowskian width iff every chord I of K splits K into two compact convex sets K₁ and K₂ such that I is a Minkowskian double normal of K₁ or K₂. Furthermore, the Euclidean version of this theorem yields a new characterization of d-dimensional Euclidean ball where d 3.     

Intrinsic characterizations for complex hypercylinders and complex hyperspheres

In terms of conditions on the curvature tensors of Riemann-Christoffel, Ricci, Weyl and Bochner we obtain several new characterizations of complex hyperspheres in complex projective spaces, of complex hypercylinders in complex Euclidean spaces and of complex hyperlanes in complex space forms.Aspirant N.F.W.O. (België).     

Elementary constructions of locally compact affine planes

In this paper we describe several elementary constructions of 4-, 8- and 16-dimensional locally compact affine planes. The new planes share many properties with the classical ones and are very easy to handle. Among the new planes we find translation planes, planes that are constructed by gluing together two halves of different translation planes, 4-dimensional shift planes, etc. We discuss various applications of our constructions, e.g. the construction of 8- and 16-dimensional affine planes with a point-transitive collineation group which are neither translation planes nor dual translation planes, the proof that a 2-dimensional affine plane that can be coordinatized by a linear ternary field with continuous ternary operation can be embedded in 4-, 8- and 16-dimensional planes, the construction of 4-dimensional non-classical planes that admit at the same time orthogonal and non-orthogonal polarities. We also consider which of our planes have tangent translation planes in all their points. In a final section we generalize the Knarr-Weigand criterion for topological ternary fields.This research was supported by a Feodor Lynen fellowship.     

Equiframed Curves – A Generalization of Radon Curves

Equiframed curves are centrally symmetric convex closed planar curves that are touched at each of their points by some circumscribed parallelogram of smallest area. These curves and their higher-dimensional analogues were introduced by Peczynski and Szarek (1991, Math Proc Cambridge Philos Soc 109: 125–148). Radon curves form a proper subclass of this class of curves. Our main result is a construction of an arbitrary equiframed curve by appropriately modifying a Radon curve. We give characterizations of each type of curve to highlight the subtle difference between equiframed and Radon curves and show that, in some sense, equiframed curves behave dually to Radon curves.Research supported by a grant from a cooperation between the Deutsche Forschungsgemeinschaft in Germany and the National Research Foundation in South Africa. Parts of this paper were written during a visit to the Department of Mathematics, Applied Mathematics and Astronomy of the University of South Africa.     

Some remarkable lines of triangles in real normed spaces and characterizations of inner product structures

Summary In this work we consider the heights and the bisectrices of a triangle in a real normed space. Using well-known formulas which can be generalized to real normed spaces we obtain a collection of new characterizations of inner product spaces.     

Polynomials and spatial Pick-type theorems

Pick's theorem about the area of a simple lattice planar polygon has many extensions and generalizations even in the planar case. The theorem has also higher-dimensional generalizations, which are not as commonly known as the 2-dimensional case. The aim of the paper is, on one hand, to give a few new higher-dimensional generalizations of Pick's theorem and, on the other hand, collect known ones. We also study some relationships between lattice points in a lattice polyhedron which lead to some new Pick-type formulae. Another purpose of this paper is to pose several problems related to the subject of higher-dimensional Pick-type theorems. We hope that the paper may popularize the idea of determining the volume of a lattice polyhedron P by reading information contained in a lattice and the tiling of the space generated by the lattice.     

A new construction of Radon curves and related topics

We present a new construction of Radon curves which only uses convexity methods. In other words, it does not rely on an auxiliary Euclidean background metric (as in the classical works of J. Radon, W. Blaschke, G. Birkhoff, and M. M. Day), and also it does not use typical methods from plane Minkowski Geometry (as proposed by H. Martini and K. J. Swanepoel). We also discuss some properties of normed planes whose unit circle is a Radon curve and give characterizations of Radon curves only in terms of Convex Geometry.     

Harmonic symmetrization of convex sets and of Finsler structures,with applications to Hilbert geometry

David Hilbert discovered in 1895 an important metric that is canonically associated to an arbitrary convex domain Ω$\Omega$ in the Euclidean (or projective) space. This metric is known to be Finslerian, and the usual proof of this fact assumes a certain degree of smoothness of the boundary of Ω$\Omega$, and refers to a theorem by Busemann and Mayer that produces the norm of a tangent vector from the distance function. In this paper, we develop a new approach for the study of the Hilbert metric where no differentiability is assumed. The approach exhibits the Hilbert metric on a domain as a symmetrization of a natural weak metric, known as the Funk metric. The Funk metric is described as a tautological   weak Finsler metric, in which the unit ball in each tangent space is naturally identified with the domain Ω$\Omega$ itself. The Hilbert metric is then identified with the reversible tautological weak Finsler structure   on Ω$\Omega$, and the unit ball of the Hilbert metric at each point is described as the harmonic symmetrization of the unit ball of the Funk metric. Properties of the Hilbert metric then follow from general properties of harmonic symmetrizations of weak Finsler structures.     

On the number of solutions to the discrete two-dimensional L0-Minkowski problem

Our main result shows the uniqueness of planar convex polygons of given outer orientations to the sides and prescribed areas of the triangles formed by the origin with any two consecutive vertices, if they exist. Such polygons are solutions to the discrete L₀-Minkowski problem in the plane proposed, in a larger generality, by Lutwak in his extension of the Brunn-Minkowski theory. The method exploits the equivalence between the L₀-polygons and the homothetic solutions to appropriate anisotropic crystalline flows. Hence as a by-product we obtain a positive answer to Jean Taylor's conjecture on the uniqueness of polygonal shapes which are deformed homothetically under the flow by crystalline curvature, without any symmetry assumptions.     

Translation Generalized Quadrangles In Even Characteristic

In this paper, we first introduce new objects called “translation generalized ovals” and “translation generalized ovoids”, and make a thorough study of these objects. We then obtain numerous new characterizations of the of Tits and the classical generalized quadrangle in even characteristic, including the complete classification of 2-transitive generalized ovals for the even case. Next, we prove a new strong characterization theorem for the of Tits. As a corollary, we obtain a purely geometric proof of a theorem of Johnson on semifield flocks. * The second author is a Postdoctoral Fellow of the Fund for Scientific Research—Flanders (Belgium).