On Reduced Triangles in Normed Planes

A convex body is reduced if it does not properly contain a convex body of the same minimal width. In this paper we present new results on reduced triangles in normed (or Minkowski) planes, clearly showing how basic seemingly elementary notions from Euclidean geometry (like that of the regular triangle) spread when we extend them to arbitrary normed planes. Via the concept of anti-norms, we study the rich geometry of reduced triangles for arbitrary norms giving bounds on their side-lengths and on their vertex norms. We derive results on the existence and uniqueness of reduced triangles, and also we obtain characterizations of the Euclidean norm by means of reduced triangles. In the introductory part we discuss different topics from Banach Space Theory, Discrete Geometry, and Location Science which, unexpectedly, benefit from results on reduced triangles.     

The smallest enclosing ball problem and the smallest intersecting ball problem: existence and uniqueness of solutions

In this paper we study the following problems and their further generalizations: given a finite number of nonempty closed subsets of a normed space, find a ball with the smallest radius that encloses all of the sets, and find a ball with the smallest radius that intersects all of the sets. These problems can be viewed as generalized versions of the smallest enclosing circle problem introduced in the nineteenth century by Sylvester which asks for the circle of smallest radius enclosing a given set of finite points in the plane. We will focus on sufficient conditions for the existence and uniqueness of an optimal solution for each problem, while the study of optimality conditions and numerical implementation will be addressed in our next projects.     

Isoperimetric triangular enclosures with a fixed angle

In this paper, we study four variants of the famous isoperimetric problem. Given a set S of n > 2 points in the plane (in general position), we show how to compute in O(n ²) time, a triangle with maximum (or minimum) area enclosing S among all enclosing triangles with fixed perimeter and one fixed angle. We also show how to compute in O(n ²) time, a triangle with maximum (or minimum) perimeter enclosing S among all enclosing triangles with fixed area and one fixed angle. We also provide an Ω (n log n) lower bound for these problems in the algebraic computation tree model.     

An optimal algorithm for finding minimal enclosing triangles

Klee and Laskowski's O(n log²n) algorithm for finding all minimal area triangles enclosing a given convex polygon of n vertices is improved to Θ(n), which is shown to be optimal both for finding all minima and for finding just one.     

Minimal Time Functions and the Smallest Intersecting Ball Problem with Unbounded Dynamics

The smallest enclosing circle problem introduced in the nineteenth century by Sylvester asks for the circle of smallest radius enclosing a given set of finite points in the plane. An extension of this problem, called the smallest intersecting ball problem, was also considered recently: given a finite number of nonempty closed subsets of a normed space, find a ball with the smallest radius that intersects all of the sets. In this paper, we initiate the study of minimal time functions generated by unbounded dynamics and discuss their applications to further extensions of the smallest enclosing circle problem. This approach continues our effort in applying convex and nonsmooth analysis to the well-established field of facility location.     

Concurrency of the altitudes of a triangle

Of all the traditional (or Greek) centers of a triangle, the orthocenter (i.e., the point of concurrence of the altitudes) is probably the one that attracted the most of attention. This may be due to the fact that it is the only one that has no exact analogue for arbitrary higher dimensional simplices, for spherical and hyperbolic triangles, or for triangles in normed planes. But it possibly has to do also with the non-existence of any explicit treatment of this center in the Greek works that have come down to us. In this paper we present different proofs of the fact that the altitudes of a triangle are concurrent. These include the first extant proof, in the works of al-Kūhī, Newton’s proof, Gauss’s proof, and other interesting proofs.     

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.     

On the stability of normalized Powell–Sabin B-splines

In this paper we show that the normalized Powell–Sabin B-splines form a stable basis for the max norm. The approximation constants depend only on the smallest angle in the underlying triangulation. Since the B-splines refer to the size of the Powell–Sabin triangles, we find that small Powell–Sabin triangles yield better approximation constants than big Powell–Sabin triangles. Next, in addition to the max norm, we treat the L_p norm. Here the approximation constants depend also on a fraction proper to the triangulation, thus the B-splines are not stable for the L_p norm. Finally, as a special case, we consider the B-spline bases obtained from Powell–Sabin triangles with minimal area and pay extra attention to the approximation constants for the max norm.     

A new type of orthogonality for normed planes

In this paper we introduce a new type of orthogonality for real normed planes which coincides with usual orthogonality in the Euclidean situation. With the help of this type of orthogonality we derive several characterizations of the Euclidean plane among all normed planes, all of them yielding also characteristic properties of inner product spaces among real normed linear spaces of dimensions d ⩾ 3.     

Triangle Contact Representations and Duality

A?contact representation by triangles of a graph is a set of triangles in the plane such that two triangles intersect on at most one point, each triangle represents a vertex of the graph and two triangles intersects if and only if their corresponding vertices are adjacent. De Fraysseix, Ossona de Mendez and Rosenstiehl proved that every planar graph admits a contact representation by triangles. We strengthen this in terms of a simultaneous contact representation by triangles of a planar map and of its dual. A?primal?Cdual contact representation by triangles of a planar map is a contact representation by triangles of the primal and a contact representation by triangles of the dual such that for every edge uv, bordering faces f and g, the intersection between the triangles corresponding to u and v is the same point as the intersection between the triangles corresponding to f and g. We prove that every 3-connected planar map admits a primal?Cdual contact representation by triangles. Moreover, the interiors of the triangles form a tiling of the triangle corresponding to the outer face and each contact point is a corner of exactly three triangles. Then we show that these representations are in one-to-one correspondence with generalized Schnyder woods defined by Felsner for 3-connected planar maps.     

Bounded and unbounded polynomials and multilinear forms: Characterizing continuity

In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the question as to whether a polynomial is continuous if and only if it transforms connected sets into connected sets. These results motivate the natural question as to how many non-continuous polynomials there are on an infinite dimensional normed space. A problem on the lineability of the sets of non-continuous polynomials and multilinear mappings on infinite dimensional normed spaces is answered.     

Tilings of Convex Polygons with Congruent Triangles

By the spectrum of a polygon A we mean the set of triples (??,??,??) such that A can be dissected into congruent triangles of angles ??,??,??. We propose a technique for finding the spectrum of every convex polygon. Our method is based on the following classification. A tiling is called regular if there are two angles of the triangles, ?? and ?? such that at every vertex of the tiling the number of triangles having angle ?? equals the number of triangles having angle ??. Otherwise the tiling is irregular. We list all pairs (A,T) such that A is a convex polygon and T is a triangle that tiles A regularly. The list of triangles tiling A irregularly is always finite, and can be obtained, at least in principle, by considering the system of equations satisfied by the angles, examining the conjugate tilings, and comparing the sides and the area of the triangles to those of A. Using this method we characterize the convex polygons with infinite spectrum, and determine the spectrum of the regular triangle, the square, all rectangles, and the regular N-gons with N large enough.     

Subdifferentials of a minimal time function in normed spaces

In a general normed vector space, we study the minimal time function determined by a differential inclusion where the set-valued mapping involved has constant values of a bounded closed convex set U and by a closed target set S. We show that proximal and Fréchet subdifferentials of a minimal time function are representable by virtue of corresponding normal cones of sublevel sets of the function and level or suplevel sets of the support function of U. The known results in the literature require the set U to have the origin as an interior point or U be compact. (In particular, if the set U is the unit closed ball, the results obtained reduce to the subdifferential of the distance function defined by S.)     

A spectrum result on minimal blocking sets with respect to the planes of PG(3, q), q odd

This article presents a spectrum result on minimal blocking sets with respect to the planes of PG(3, q), q odd. We prove that for every integer k in an interval of, roughly, size [q ²/4, 3q ²/4], there exists such a minimal blocking set of size k in PG(3, q), q odd. A similar result on the spectrum of minimal blocking sets with respect to the planes of PG(3, q), q even, was presented in Rößing and Storme (Eur J Combin 31:349–361, 2010). Since minimal blocking sets with respect to the planes in PG(3, q) are tangency sets, they define maximal partial 1-systems on the Klein quadric Q ⁺(5, q), so we get the same spectrum result for maximal partial 1-systems of lines on the Klein quadric Q ⁺(5, q), q odd.     

Concepts of curvatures in normed planes

The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a comprehensive overview on geometric properties of and relations between all introduced curvature concepts, we try to fill this gap. To complete and clarify the whole picture, we show which known concepts are equivalent, and add also a new type of curvature. Certainly, this yields a basis for further research and also for possible extensions of the whole existing framework. In addition, we derive various new results referring in full broadness to the variety of known curvature types in normed planes. These new results involve characterizations of curves of constant curvature, new characterizations of Radon planes and the Euclidean subcase, and analogues to classical statements like the four vertex theorem and the fundamental theorem on planar curves. We also introduce a new curvature type, for which we verify corresponding properties. As applications of the little theory developed in our expository paper, we study the curvature behavior of curves of constant width and obtain also new results on notions like evolutes, involutes, and parallel curves.     

Geometric mean and triangles inscribed in a semicircle in Banach spaces

We consider the triangles with vertices x, −x and y where x,y are points on the unit sphere of a normed space. Using the geometric means of the variable lengths of the sides of these triangles, we define two geometric constants for Banach spaces. These constants are closely related to the modulus of convexity of the space under consideration, and they seem to represent a useful tool to estimate the exact values of the James and Jordan–von Neumann constants of some Banach spaces.     

Cassini Curves in Normed Planes

We extend the concept of Cassini curves from the Euclidean plane to normed (or Minkowski) planes and show that geometric properties of (Minkowskian) Cassini curves are closely related to geometric properties of the unit disc determining the underlying normed plane.     

On the Napoleon-Torricelli configuration in Affine Cayley-Klein planes

There are three affine Cayley-Klein planes (see [5]), namely, the Euclidean plane, the isotropic (Galilean) plane, and the pseudo-Euclidean (Minkow-skian or Lorentzian) plane. We extend the generalization of the well-known Napoleon theorem related to similar triangles erected on the sides of an arbitrary triangle in the Euclidean plane to all affine Cayley-Klein planes. Using the Ω_k-and anti-Ω_k-equilateral triangles introduced in [28], we construct the Napoleon and the Torricelli triangle of an arbitrary triangle in any affine Cayley-Klein plane. Some interesting geometric properties of these triangles are derived. The author is partially supported by grant VU-MI-204/2006.