Bounds for solid angles of lattices of rank three

We find sharp absolute constants C₁ and C₂ with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval [C₁,C₂]. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we derive a lower bound in terms of the ratios of successive minima of the lattice. We use this result to show that among all spherical triangles on the unit sphere in RN with vertices on the minimal vectors of a lattice, the smallest possible area is achieved by a configuration of minimal vectors of the (normalized) face centered cubic lattice in R³. Such spherical configurations come up in connection with the kissing number problem.     

Dissections of regular polygons into triangles of equal areas

This paper answers the question, “If a regular polygon withn sides is dissected intom triangles of equal areas, mustm be a multiple ofn?” Forn=3 the answer is “no,” since a triangle can be cut into any positive integral number of triangles of equal areas. Forn=4 the answer is again “no,” since a square can be cut into two triangles of equal areas. However, Monsky showed that a square cannot be dissected into an odd number of triangles of equal areas. We show that ifn is at least 5, then the answer is “yes.” Our approach incorporates the techniques of Thomas, Monsky, and Mead, in particular, the use of Sperner's lemma and non-Archimedean valuations, but also makes use of affine transformations to distort a given regular polygon into one to which those techniques apply.     

Affine diffeomorphisms of translation surfaces: Periodic points, Fuchsian groups, and arithmeticity

We study translation surfaces with rich groups of affine diffeomorphisms—“prelattice” surfaces. These include the lattice translation surfaces studied by W. Veech. We show that there exist prelattice but nonlattice translation surfaces. We characterize arithmetic surfaces among prelattice surfaces by the infinite cardinality of their set of points periodic under affine diffeomorphisms. We give examples of translation surfaces whose periodic points and Weierstrass points coincide.     

Affine rotational surfaces with vanishing affine curvature

We give a classification of affine rotational surfaces in affine 3-space with vanishing affine Gauss-Kronecker curvature. Non-degenerated surfaces in three dimensional affine space with affine rotational symmetry have been studied by a number of authors (I.C. Lee. [3], P. Lehebel [4], P.A. Schirokow [10], B. Su [12], W. Süss [13]). In the present paper we study these surfaces with the additional property of vanishing affine Gauss-Kronecker curvature, that means the determinant of the affine shape operator is zero. We give a complete classification of these surfaces, which are the affine analogues to the cylinders and cones of rotation in euclidean geometry. These surfaces are examples of surfaces with diagonalizable rank one (affine) shape operator (cf. B. Opozda [8] and B. Opozda, T. Sasaki [7]). The affine normal images are curves.     

On triangular billiards

We prove a conjecture of Kenyon and Smillie concerning the nonexistence of acute rational-angled triangles with the lattice property.     

Pants Decompositions of Random Surfaces

Our goal is to show, in two different contexts, that “random” surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus g for which any pants decomposition requires curves of total length at least g ^7/6−ε . Moreover, we prove that this bound holds for most metrics in the moduli space of hyperbolic metrics equipped with the Weil–Petersson volume form. We then consider surfaces obtained by randomly gluing euclidean triangles (with unit side length) together and show that these surfaces have the same property.     

On the number of hamiltonian cycles in triangulations with few separating triangles

In this article, we investigate the number of hamiltonian cycles in triangulations. We improve a lower bound of for the number of hamiltonian cycles in triangulations without separating triangles (4‐connected triangulations) by Hakimi, Schmeichel, and Thomassen to a linear lower bound and show that a linear lower bound even holds in the case of triangulations with one separating triangle. We confirm their conjecture about the number of hamiltonian cycles in triangulations without separating triangles for up to 25 vertices and give computational results and constructions for triangulations with a small number of hamiltonian cycles and 1–5 separating triangles.     

Ricci curvature and betti numbers

We derive a uniform bound for the total betti number of a closed manifold in terms of a Ricci curvature lower bound, a conjugate radius lower bound and a diameter upper bound. The result is based on an angle version of Toponogov comparison estimate for small triangles in a complete manifold with a Ricci curvature lower bound. We also give a uniform estimate on the generators of the fundamental group and prove a fibration theorem in this setting.     

An improvement of the Feng–Rao bound on minimum distance

We present a lower bound for the minimum distance of certain affine variety codes which is better than the Feng–Rao bound. We also introduce a minimum distance estimate which, at times, can serve as a lower bound as well.     

On quotients of Coxeter groups under the weak order

We consider quotients of finitely generated Coxeter groups under the weak order. Björner and Wachs proved that every such quotient is a meet semi-lattice, and in the finite case is a lattice [Björner and Wachs, Trans. Amer. Math. Soc. 308 (1988) 1–37]. Our result is that the quotient of an affine Weyl group by the corresponding finite Weyl group is a lattice, and that up to isomorphism, these are the only quotients of infinite Coxeter groups that are lattices. In this paper, we restrict our attention to the non-affine case; the affine case appears in [Waugh, Order 16 (1999) 77–87]. We reduce to the hyperbolic case by an argument using induced subgraphs of Coxeter graphs. Within each quotient, we produce a set of elements with no common upper bound, generated by a Maple program. The number of cases is reduced because the sets satisfy the following conjecture: if a set of elements does not have an upper bound in a particular Coxeter group, then it does not have an upper bound in any Coxeter group whose graph can be obtained from the graph of the original group by increasing edge weights.     

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.     

Area functions on affine planes

Oriented area functions are functions defined on the set of ordered triangles of an affine plane which are antisymmetric under odd permutations of the vertices and which behave additively when triangles are cut into two. We compare several elementary properties which such an area function may have (roughly speaking shear invariance, equality of area of the two triangles obtained by cutting a parallelogram along a diagonal, and equality of area of the two triangles obtained by cutting a triangle along a median). It turns out purely by arguments of elementary affine geometry (if cleverly arranged) that these properties are grosso modo equivalent, although one has to be careful about “pathological” situations. Furthermore, all oriented area functions satisfying these properties are explicitly determined. Finally they are compared with so-called geometric valuations.     

On the maximal number of exceptional values of Gauss maps for various classes of surfaces

The main goal of this paper is to reveal the geometric meaning of the maximal number of exceptional values of Gauss maps for several classes of immersed surfaces in space forms, for example, complete minimal surfaces in the Euclidean three-space, weakly complete improper affine spheres in the affine three-space and weakly complete flat surfaces in the hyperbolic three-space. For this purpose, we give an effective curvature bound for a specified conformal metric on an open Riemann surface.     

Die Flächen mit geraden Linien als Affinen Zentraflächen

If the (euclidean) Gauss curvature of a surface in 3-space is nowhere vanishing, we have a uniqueaffine normal in every point of the surface and the set of affine normals forms a line congruence. According to the euclidean situation one can discuss existence and degeneration ofaffine focal surfaces. A cylinder of revolution in euclidean space can be characterized by the property, that all normals intersecttwo straight lines. The corresponding property in affine geometry leads to certainaffine surfaces of revolution, whose meridian curves (plane sections through a fixed axis) can be determined. If we assume the affine focal surfaces to coincide (the affine Weingarten endomorphism has double eigenvalues in every point) and degenerate intoone straight line, then the surface is aruled surface the generators of which areparallel to a plane.

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Herrn Professor Dr. Gerhard Geise zum 70. Geburtstag gewidmet     

凸体的曲率映象与仿射表面积

本文研究了一些特殊凸体与其极体的曲率仿射表面积乘积的下界.对任意两个凸体,建立了它们的投影体的混合体积与其仿射表面积的一个不等式（见文[1-15]）.     

Mixed-integer sets from two rows of two adjacent simplex bases

In Andersen et al. (Lecture Notes in Computer Science, vol. 4513, Springer, Berlin, pp. 1–15, 2007) we studied a mixed-integer set arising from two rows of a simplex tableau. We showed that facets of such a set can be obtained from lattice point free triangles and quadrilaterals associated with either three or four variables. In this paper we generalize our findings and show that, when upper bounds on the non-basic variables are also considered, further classes of facets arise that cannot be obtained from triangles and quadrilaterals. Specifically, when exactly one upper bound on a non-basic variable is introduced, stronger inequalities that can be derived from pentagons involving up to six variables also appear.     

On the approximation of a smooth surface with a triangulated mesh

We approximate the normals and the area of a smooth surface with the normals and the area of a triangulated mesh whose vertices belong to the smooth surface. Both approximations only depend on the triangulated mesh (which is supposed to be known), on an upper bound on the smooth surface's curvature, on an upper bound on its reach (which is linked to the local feature size) and on an upper bound on the Hausdorff distance between both surfaces.

We show in particular that the upper bound on the error of the normals is better when triangles are right-angled (even if there are small angles). We do not need every angle to be quite large. We just need each triangle of the triangulated mesh to contain at least one angle whose sinus is large enough.     

A sextic holomorphic form of affine surfaces with constant affine mean curvature

We extend an original idea of Calabi for affine maximal surfaces and define a sextic holomorphic differential form for affine surfaces with constant affine mean curvature. We get some rigidity results for affine complete surfaces by using this sextic holomorphic form. Received: 17 May 2003     

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.