Band 11, Heft 4

We prove a quadrilateral comparison result for convex surfaces, involving projections onto quasigeodesics. As an application, we locate the set of all farthest points (i.e., points at maximal intrinsic distance from some point) in a convex surface, with respect to a simple closed quasigeodesic.

We find the minimal value of the length in de Sitter space of closed space-like curves with non-vanishing non-space-like geodesic curvature vector. These curves are in correspondence with closed almost-regular canal surfaces, and their length is a natural magnitude in conformal geometry. As an application, we get a lower bound for the total conformal torsion of closed space curves.

Given an irreducible conic in PG(2, q ) with q odd, how many points of another irreducible conic 𝒟 in PG(2, q ) are external to ? It is not hard to find 𝒟 such that either all its points off are external to , or none of its points are. Apart from these cases, we prove that for q large enough, the number we seek differs from by at most .

We introduce a new parameter, symmetric average, which measures the asymmetry of a given non-degenerated convex body K in . Namely, sav ( K ) = inf a ∈int K ∫ K a ‖– x ‖ K a dx /| K |, where | K | denotes the volume of K and K a = K – a . We show that for polytopes sav ( K ) ≤ C ln N , where N is the number of facets of K . Moreover, in general and equality in the lower bound holds if and only if K is centrally symmetric. We apply these estimates to provide bounds for covering K by homothets of K ∩ – K .

Let ℰ be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X of dimension n having a global section whose zero locus Z is a smooth subvariety of dimension n – r ≥ 3 of X . Let H be an ample line bundle on X such that its restriction H Z to Z is generated by global sections. The structure of triplets ( X , ℰ, H ) is determined under the assumption that ( Z, H Z ) has sectional genus g ( Z, H Z ) = 3.

Kleinewillinghöfer classified Laguerre planes with respect to central automorphisms . Polster and Steinke investigated flat Laguerre planes and their so-called Kleinewillinghöfer types, that is, the Kleinewillinghöfer types with respect to the full automorphism group. For some of these types the existence question remained open. We provide strong necessary existence conditions for flat Laguerre planes of Kleinewillinghöfer type III.B and provide examples of such planes of types III.B.1 and III.B.3.

In [Bini, van Geemen, Kelly, Mirror quintics, discrete symmetries and Shioda maps, 2009] some quotients of one-parameter families of Calabi–Yau varieties are related to the family of Mirror Quintics by using a construction due to Shioda. In this paper, we generalize this construction to a wider class of varieties. More specifically, let A be an invertible matrix with non-negative integer entries. We introduce varieties X A and in weighted projective space and in , respectively. The variety turns out to be a quotient of a Fermat variety by a finite group. As a by-product, X A is a quotient of a Fermat variety and is a quotient of X A by a finite group. We apply this construction to some families of Calabi–Yau manifolds in order to show their birationality.

We investigate functional properties of the mixed volume that are sufficient to characterize it, up to elementary modifications.

For d = 2, 3 a generic convex body in has a unique lattice packing and for d ≥ 4 at most a ( d ) lattice packings of maximum density, where a ( d ) ≥ 1 is a constant. If in a certain connectedness property holds, one may take a ( d ) = 1. Dually, for d = 2 a generic convex body has a unique lattice covering of minimum density and for d ≥ 3 there is a constant b ( d ) ≥ 1 such that it has at most b ( d ) such coverings.

In 1980, Arnold studied the classification problem for convex lattice polygons of given area. Since then this problem and its high dimensional analogue have been studied by Bárány, Pach, Vershik and others. Bounds for the number of non-equivalent d -dimensional convex lattice polytopes of given volume have been achieved. In this paper we study Arnold's problem for centrally symmetric lattice polygons and the classification problem for convex lattice polytopes of given cardinality. In the plane we obtain analogues to the bounds of Arnold, Bárány and Pach in both cases. However, the number of non-equivalent d -dimensional convex lattice polytopes of w lattice points is infinite whenever w – 1 ≥ d ≥ 3, which may intuitively contradict to Bárány and Vershik's upper bound.

A classical result of Blichfeldt, from 1921, gives a sharp lower bound on the volume of a convex body K , whose lattice points span the whole space, in terms of the lattice point enumerator . We are interested in a version of this inequality on the set of 0-symmetric convex bodies. Our motivation to study this problem comes from a lack of methods that exploit the symmetry assumption in problems of a similar kind and where 0-symmetry is a natural condition. We report upon sharp Blichfeldt-type inequalities for 0-symmetric lattice polygons, lattice crosspolytopes and lattice zonotopes.

A short survey is given on classification results for valuations on function spaces. Real valued, matrix valued and convex body valued valuations on Lebesgue spaces and on Sobolev spaces are considered.