Hyperelliptic Jacobians as Billiard Algebra of Pencils of Quadrics: Beyond Poncelet Porisms

The thirty years old programme of Griffiths and Harris of understanding higher-dimensional analogues of Poncelet-type problems and synthetic approach to higher genera addition theorems has been settled and completed in this paper. Starting with the observation of the billiard nature of some classical constructions and configurations, we construct the billiard algebra, that is a group structure on the set T of lines in $R^d$ simultaneously tangent to d-1 quadrics from a given confocal family. Using this tool, the related results of Reid, Donagi and Knoerrer are further developed, realized and simplified. We derive a fundamental property of T: any two lines from this set can be obtained from each other by at most d-1 billiard reflections at some quadrics from the confocal family. We introduce two hierarchies of notions: s-skew lines in T and s-weak Poncelet trajectories, s = -1,0,...,d-2. The interrelations between billiard dynamics, linear subspaces of intersections of quadrics and hyperelliptic Jacobians developed in this paper enabled us to obtain higher-dimensional and higher-genera generalizations of several classical genus 1 results: the Cayley's theorem, the Weyr's theorem, the Griffiths-Harris theorem and the Darboux theorem.Comment: 36 pages, 11 figures; to be published in Advances in Mathematic

On mm-point homogeneous polytopes in Euclidean spaces

This paper is devoted to the study the $m$-point homogeneity property and the point homogeneity degree for finite metric spaces. Since the vertex sets of regular polytopes, as well as of some their generalizations, are homogeneous, we pay much attention to the study of the homogeneity properties of the vertex sets of polytopes in Euclidean spaces. Among main results, there is a classification of polyhedra with all edges of equal length and with 2-point homogeneous vertex sets. In addition, a significant part of the paper is devoted to the development of methods and tools for studying the relevant objects.Comment: 24 page

Ball and Spindle Convexity with respect to a Convex Body

Let $C\subset {\mathbb R}^n$ be a convex body. We introduce two notions of convexity associated to C. A set $K$ is $C$-ball convex if it is the intersection of translates of $C$, or it is either $\emptyset$, or ${\mathbb R}^n$. The $C$-ball convex hull of two points is called a $C$-spindle. $K$ is $C$-spindle convex if it contains the $C$-spindle of any pair of its points. We investigate how some fundamental properties of conventional convex sets can be adapted to $C$-spindle convex and $C$-ball convex sets. We study separation properties and Carath\'eodory numbers of these two convexity structures. We investigate the basic properties of arc-distance, a quantity defined by a centrally symmetric planar disc $C$, which is the length of an arc of a translate of $C$, measured in the $C$-norm, that connects two points. Then we characterize those $n$-dimensional convex bodies $C$ for which every $C$-ball convex set is the $C$-ball convex hull of finitely many points. Finally, we obtain a stability result concerning covering numbers of some $C$-ball convex sets, and diametrically maximal sets in $n$-dimensional Minkowski spaces.Comment: 27 pages, 5 figure