Inscribing a symmetric body in an ellipse

We prove that any bounded, centrally symmetric object K in the plane can be inscribed in an ellipse E touching its boundary aK at at least four points. An application to Minkowskian geometry is given

Neighborly inscribed polytopes and Delaunay triangulations

We construct a large family of neighborly polytopes that can be realized with all the vertices on the boundary of any smooth strictly convex body. In particular, we show that there are superexponentially many combinatorially distinct neighborly polytopes that admit realizations inscribed on the sphere. These are the first examples of inscribable neighborly polytopes that are not cyclic polytopes, and provide the current best lower bound for the number of combinatorial types of inscribable polytopes (which coincides with the current best lower bound for the number of combinatorial types of polytopes). Via stereographic projections, this translates into a superexponential lower bound for the number of combinatorial types of (neighborly) Delaunay triangulations.Comment: 15 pages, 2 figures. We extended our results to arbitrary smooth strictly convex bodie

Espaces de splines réproduisant les polynômes à partir de pavages de zonotopes

Revised version March 2021Given a point configuration A, we uncover a connection between polynomial-reproducing spline spaces over subsets of conv(A) and fine zonotopal tilings of the zonotope Z(V) associated to the corresponding vector configuration. This link directly generalizes a known result on Delaunay configurations and naturally encompasses, due to its combinatorial character, the case of repeated and affinely dependent points in A. We prove the existence of a general iterative construction process for such spaces. Finally, we turn our attention to regular fine zonotopal tilings, specializing our previous results and exploiting the adjacency graph of the tiling to propose a set of practical algorithms for the construction and evaluation of the associated spline functions.Étant donné une configuration de points A, on explore une connexion entre les espaces de splines reproduisant les polynômes sur certains sous-ensembles de conv(A) et les pavages fins du zonotope Z(V) associé à la configuration de vecteurs correspondante. Ce lien généralise directement un résultat connu sur les configurations de Delaunay et inclut naturellement, grâce à son charactère combinatoire, le cas de points en répétés et affinement dépendants en A. On prouve l'existence d'un processus de construction itératif général pour ces espaces. Enfin, on tourne notre attention vers les pavages de zonotopes fins et réguliers, en spécialisant nos résultats précédentes et en exploitant le graphe d'adjacence du pavage afin de proposer un ensemble d'algorithmes utiles en pratique pour la construction et l'évaluation des fonctions splines associées

Aspects of Metric Spaces in Computation

Metric spaces, which generalise the properties of commonly-encountered physical and abstract spaces into a mathematical framework, frequently occur in computer science applications. Three major kinds of questions about metric spaces are considered here: the intrinsic dimensionality of a distribution, the maximum number of distance permutations, and the difficulty of reverse similarity search. Intrinsic dimensionality measures the tendency for points to be equidistant, which is diagnostic of high-dimensional spaces. Distance permutations describe the order in which a set of fixed sites appears while moving away from a chosen point; the number of distinct permutations determines the amount of storage space required by some kinds of indexing data structure. Reverse similarity search problems are constraint satisfaction problems derived from distance-based index structures. Their difficulty reveals details of the structure of the space. Theoretical and experimental results are given for these three questions in a wide range of metric spaces, with commentary on the consequences for computer science applications and additional related results where appropriate

On Delaunay Oriented Matroids

The fact that for any finite set S of points in the Euclidean plane E² one can define an oriented matroid in terms of how spheres partition it is well-known and easy to proof via the lifting property of Delaunay triangulations (cf. [1] and [2]) . Here we give a new definition of these oriented matroids (that we call Delaunay oriented matroids) which trivially generalizes to arbitrary dimension and explicitly show the precise relation of Delaunay oriented matroids with not only the usual Voronoi diagrams and Delaunay triangulations, but with Voronoi diagrams of arbitrary order k. Moreover we show that the existence of these Delaunay oriented matroids is not really dependent on the lifting property of Delaunay triangulations but on some nice properties of Euclidean spheres. In fact, we generalize the definition to smooth, strictly convex distances in the plane (cf. x2) which have not the lifting property as we also show and to any metrics whose "spheres" have some nice properties (..