TY - JOUR AB - Voronoi diagrams and Delaunay triangulations have been extensively used to represent and compute geometric features of point configurations. We introduce a generalization to poset diagrams and poset complexes, which contain order-k and degree-k Voronoi diagrams and their duals as special cases. Extending a result of Aurenhammer from 1990, we show how to construct poset diagrams as weighted Voronoi diagrams of average balls. AU - Edelsbrunner, Herbert AU - Iglesias Ham, Mabel ID - 1295 JF - Electronic Notes in Discrete Mathematics TI - Multiple covers with balls II: Weighted averages VL - 54 ER - TY - JOUR AB - We give explicit formulas and algorithms for the computation of the Thurston–Bennequin invariant of a nullhomologous Legendrian knot on a page of a contact open book and on Heegaard surfaces in convex position. Furthermore, we extend the results to rationally nullhomologous knots in arbitrary 3-manifolds. AU - Durst, Sebastian AU - Kegel, Marc AU - Klukas, Mirko D ID - 1292 IS - 2 JF - Acta Mathematica Hungarica TI - Computing the Thurston–Bennequin invariant in open books VL - 150 ER - TY - JOUR AB - In this paper we investigate the existence of closed billiard trajectories in not necessarily smooth convex bodies. In particular, we show that if a body K ⊂ Rd has the property that the tangent cone of every non-smooth point q ∉ ∂K is acute (in a certain sense), then there is a closed billiard trajectory in K. AU - Akopyan, Arseniy AU - Balitskiy, Alexey ID - 1330 IS - 2 JF - Israel Journal of Mathematics TI - Billiards in convex bodies with acute angles VL - 216 ER - TY - JOUR AB - We apply the technique of Károly Bezdek and Daniel Bezdek to study billiard trajectories in convex bodies, when the length is measured with a (possibly asymmetric) norm. We prove a lower bound for the length of the shortest closed billiard trajectory, related to the non-symmetric Mahler problem. With this technique we are able to give short and elementary proofs to some known results. AU - Akopyan, Arseniy AU - Balitskiy, Alexey AU - Karasev, Roman AU - Sharipova, Anastasia ID - 1360 IS - 10 JF - Proceedings of the American Mathematical Society TI - Elementary approach to closed billiard trajectories in asymmetric normed spaces VL - 144 ER - TY - JOUR AB - The concept of well group in a special but important case captures homological properties of the zero set of a continuous map (Formula presented.) on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within (Formula presented.) distance r from f for a given (Formula presented.). The main drawback of the approach is that the computability of well groups was shown only when (Formula presented.) or (Formula presented.). Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of (Formula presented.) by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and (Formula presented.), our approximation of the (Formula presented.)th well group is exact. For the second part, we find examples of maps (Formula presented.) with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status. AU - Franek, Peter AU - Krcál, Marek ID - 1408 IS - 1 JF - Discrete & Computational Geometry TI - On computability and triviality of well groups VL - 56 ER -