Unit disk of the smallest self-perimeter in a Minkowski plane

On a Minkowski plane with nonsymmetric metric, we find all unit disks with self-perimeter 6, the smallest possible value.     

Tilings of the plane with unit area triangles of bounded diameter

There exist tilings of the plane with pairwise noncongruent triangles of equal area and bounded perimeter. Analogously, there exist tilings with triangles of equal perimeter, the areas of which are bounded from below by a positive constant. This solves a problem of Nandakumar.     

Monochromatic triangles in two-colored plane

We prove that for any partition of the plane into a closed set C and an open set O and for any configuration T of three points, there is a translated and rotated copy of T contained in C or in O.     

Evolutes of fronts in the Minkowski plane

We consider the differential geometry of evolutes of singular curves and give the definitions of spacelike fronts and timelike fronts in the Minkowski plane. We also give the notions of moving frames along the non‐lightlike fronts in the Minkowski plane. By using the moving frames, we define the evolutes of non‐lightlike fronts and investigate the geometric properties of these evolutes. We obtain that the evolute of a spacelike front is a timelike front and the evolute of a timelike front is a spacelike front. Since the evolute of a non‐lightlike front is also a non‐lightlike front, we can take evolute again. We study the Minkowski Zigzag number of non‐lightlike fronts and give the n‐th evolute of the non‐lightlike front. Finally, we give an example to illustrate our results.     

On the perimeters of simple polygons contained in a disk

A simple n-gon is a polygon with n edges with each vertex belonging to exactly two edges and every other point belonging to at most one edge. Brass et?al. (Research Problems in Discrete Geometry, 2005) asked the following question: For n ???5 odd, what is the maximum perimeter of a simple n-gon contained in a Euclidean unit disk? In 2009, Audet et?al. (Discrete Comput Geom 41:208?C215) answered this question, and showed that the optimal configuration is an isosceles triangle with a multiple edge, inscribed in the disk. In this note we give a shorter and simpler proof of their result, which we generalize also for hyperbolic disks, and for spherical disks of sufficiently small radii.     

Cauchy-Riemann equations in Minkowski plane

The properties of the symmetry and ordering of Minkowski plane are discussed by using hyperbolic imaginary unit and elliptic imaginary unit of Clifford algebra, and the representations of Cauchy-Riemann equations are given in Minkowski plane.     

A note on tropical triangles in the plane

We define transversal tropical triangles （affine and projective） and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically independent and they tropically span a classical hexagon whose sides have slopes ∞, 0, 1. Using this classical hexagon, we determine a parameter space for transversal tropical triangles. The coordinates of the vertices of a transversal tropical triangle determine a tropically regular matrix. Triangulations of the tropical plane are obtained.     

Skeletal tilings of the plane into equal triangles

Translated from Matematicheskie Zametki, Vol. 55, No. 1, pp. 105–116, January, 1994.     

Edges not contained in triangles and the number of contractible edges in a 4-connected graph

Let G be a 4-connected graph, and let E_c(G) denote the set of 4-contractible edges of G and let denote the set of those edges of G which are not contained in a triangle. Under this notation, we show that if , then we have .     

Large regular simplices contained in a hypercube

We prove that the n-dimensional unit hypercube contains an n-dimensional regular simplex of edge length c√n, where c > 0 is a constant independent of n. Supported by MEXT Grant-in-Aid for Scientific Research (B) 20340022.     

Edges not contained in triangles and the distribution of contractible edges in a 4-connected graph

We prove results concerning the distribution of 4-contractible edges in a 4-connected graph G in connection with the edges of G not contained in a triangle. As a corollary, we show that if G is 4-regular 4-connected graph, then the number of 4-contractible edges of G is at least one half of the number of edges of G not contained in a triangle.     

A characterization of the semi-isometries of a Minkowski plane over a field K

Under certain hypotheses on the field K,W.Bens has proved that a map of the plane K², which preserves a single Lorents-Minkowski — distance is semilinear and injective [5],[6]. We shall generalise this result for every field K with char K ? {2,3,5 } and for K = GF(5^m), m> 1.     

Covering the plane with copies of a convex disk

We prove that for every convex disk in the plane there exists a double-lattice covering of the plane with copies of with density ≤ 1.2281772 . This improves the best previously known upper bound ≤ 8/(3+2\sqrt{3}) 1.2376043 , due to Kuperberg, but it is still far from the conjectured value . <lsiheader> <onlinepub>7 August, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>20n2p251.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>no <sectionname> </lsiheader> Received June 1, 1996, and in revised form January 24, 1997.     

Points and triangles in the plane and halving planes in space

We prove that for any setS ofn points in the plane andn ^3−α triangles spanned by the points inS there exists a point (not necessarily inS) contained in at leastn ^3−3α/(c log⁵ n) of the triangles. This implies that any set ofn points in three-dimensional space defines at most halving planes. Work on this paper by Boris Aronov and Rephael Wenger has been supported by DIMACS under NSF Grant STC-88-09648. Work on this paper by Bernard Chazelle has been supported by NSF Grant CCR-87-00917. Work by Herbert Edelsbrunner has been supported by NSF Grant CCR-87-14565. Micha Sharir has been supported by ONR Grant N00014-87-K-0129, by NSF Grant CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the Israeli National Council for Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences.     

The size of a Minkowski ellipse that contains the unit ball

In this paper we study the minimum radius of Minkowski ellipses (with antipodal foci on the unit sphere) necessary to contain the unit ball of a (normed or) Minkowski plane. We obtain a general upper bound depending on the modulus of convexity, and in the special case of a so-called symmetric Minkowski plane (a notion that we will recall in the paper) we prove a lower bound, and also we obtain that 3 is the exact upper bound.