Mathematics

We study betweenness preserving mappings (we call them monotone) on subsets of the plane. We show, in particular, that an open planar set cannot be mapped in a one-to-one monotone way into the real line.

We study the locus of the Circumcenter of Mass of Poncelet polygons, and the limit of the Center of Mass (when we consider the polygon as a "homogeneous lamina") for degenerate Poncelet polygons. We also provide a proof for one of Dan Reznik invariants for billiard trajectories. Plus, we take a look at how the scene looks like when we shift to spherical geometry.

This is the second part of a series of papers concerning Morse quasiflats -- higher dimensional analogs of Morse quasigeodesics. Our focus here is on their asymptotic structure. In metric spaces with convex geodesic bicombings, we prove asymptotic conicality, uniqueness of tangent cones at infinity and Euclidean volume growth rigidity. Moreover, we provide some first applications.

We analyze some properties of a class of multiexponential maps appearing naturally in the geometric analysis of Carnot groups. We will see that such maps can be useful in at least two interesting problems. First, in relation to the analysis of some regularity properties of horizontally convex sets. Then, we will show that our multiexponential maps can be used to prove the Pansu differentiability of the subRiemannian distance from a fixed point.

We show that for any n -dimensional lattice L⊆ R n , the torus R n /L can be embedded into Hilbert space with O( nlogn − − − − − − √ ) distortion. This improves the previously best known upper bound of O(n logn − − − − √ ) shown by Haviv and Regev (APPROX 2010) and approaches the lower bound of Ω( n − − √ ) due to Khot and Naor (FOCS 2005, Math. Annal. 2006).

This article extends the range of 1-DOF rigid-foldable developable quadrilateral creased papers. In a previous article, we put forward a sufficient and necessary condition for a quadrilateral creased paper to be rigid-foldable, and introduce a special sufficient condition that is convenient for practical use, generating quadrilateral creased paper by stitching basic units. In this article we develop new flat-foldable units and show how these can be used to construct a series of more complex rigid-foldable developable quadrilateral creased papers.

Starting with a finite point set X⊂ R d , the peeling process repeatedly removes the set of the vertices of the convex hull of the current set. The number of peeling steps required to completely remove X is called the layer number of X , denoted by L(X) . In the article, we study the layer number of evenly distributed families of point sets contained in B d , the d -dimensional unit ball. These sets consist of points in B d whose minimal distance is asymptotically as large as possible. We show that for a set X belonging to an evenly distributed family, L(X)≥Ω(|X | 1/d ) holds, with the bound being asymptotically sharp. On the other hand, building on earlier results, we prove that L(X)≤O(|X | 2/d ) holds for d≥2 , which improves greatly on the current upper bound of O(|X | (d+1)/2d ) for d≥3 . Finally, we provide a recursive construction of evenly distributed families whose sets satisfy L(X)=Θ(|X | 2/d−1/(d 2 d−1 ) ) , showing that our upper bound is nearly tight.

We improve the previously best known upper bounds on the sizes of θ -spherical codes for every θ< θ ∗ ≈ 62.997 ∘ at least by a factor of 0.4325 , in sufficiently high dimensions. Furthermore, for sphere packing densities in dimensions n≥2000 we have an improvement at least by a factor of 0.4325+ 51 n . Our method also breaks many non-numerical sphere packing density bounds in small dimensions. Apart from Cohn and Zhao's \cite{CohnZhao} improvement on the geometric average of Levenshtein's bound \cite{Leven79} over all sufficiently high dimensions by a factor of 0.79, our work is the first improvement for each dimension since the work of Kabatyanskii and Levenshtein \cite{KL} and its later improvement by Levenshtein \cite{Leven79}. Moreover, we generalize Levenshtein's optimal polynomials and provide explicit formulae for them that may be of independent interest. For 0<θ< θ ∗ , we construct a test function for Delsarte's linear programing problem for θ -spherical codes with exponentially improved factor in dimension compared to previous test functions.

Classical (Euclidean) Laguerre geometry studies oriented hyperplanes, oriented hyperspheres, and their oriented contact in Euclidean space. We describe how this can be generalized to arbitrary Cayley-Klein spaces, in particular hyperbolic and elliptic space, and study the corresponding groups of Laguerre transformations. We give an introduction to Lie geometry and describe how these Laguerre geometries can be obtained as subgeometries. As an application of two-dimensional Lie and Laguerre geometry we study the properties of incircular nets.

Necessary and sufficient conditions are obtained for the infinitesimal rigidity of braced grids in the plane with respect to non-Euclidean norms. Component rectangles of the grid may carry 0, 1 or 2 diagonal braces, and the combinatorial part of the conditions is given in terms of a matroid for the bicoloured bipartite multigraph defined by the braces.