TY - JOUR AB - Using the geodesic distance on the n-dimensional sphere, we study the expected radius function of the Delaunay mosaic of a random set of points. Specifically, we consider the partition of the mosaic into intervals of the radius function and determine the expected number of intervals whose radii are less than or equal to a given threshold. We find that the expectations are essentially the same as for the Poisson–Delaunay mosaic in n-dimensional Euclidean space. Assuming the points are not contained in a hemisphere, the Delaunay mosaic is isomorphic to the boundary complex of the convex hull in Rn+1, so we also get the expected number of faces of a random inscribed polytope. As proved in Antonelli et al. [Adv. in Appl. Probab. 9–12 (1977–1980)], an orthant section of the n-sphere is isometric to the standard n-simplex equipped with the Fisher information metric. It follows that the latter space has similar stochastic properties as the n-dimensional Euclidean space. Our results are therefore relevant in information geometry and in population genetics. AU - Edelsbrunner, Herbert AU - Nikitenko, Anton ID - 87 IS - 5 JF - Annals of Applied Probability TI - Random inscribed polytopes have similar radius functions as Poisson-Delaunay mosaics VL - 28 ER - TY - JOUR AB - We prove that any cyclic quadrilateral can be inscribed in any closed convex C1-curve. The smoothness condition is not required if the quadrilateral is a rectangle. AU - Akopyan, Arseniy AU - Avvakumov, Sergey ID - 6355 JF - Forum of Mathematics, Sigma SN - 2050-5094 TI - Any cyclic quadrilateral can be inscribed in any closed convex smooth curve VL - 6 ER - TY - JOUR AB - In 1945, A.W. Goodman and R.E. Goodman proved the following conjecture by P. Erdős: Given a family of (round) disks of radii r1, … , rn in the plane, it is always possible to cover them by a disk of radius R= ∑ ri, provided they cannot be separated into two subfamilies by a straight line disjoint from the disks. In this note we show that essentially the same idea may work for different analogues and generalizations of their result. In particular, we prove the following: Given a family of positive homothetic copies of a fixed convex body K⊂ Rd with homothety coefficients τ1, … , τn> 0 , it is always possible to cover them by a translate of d+12(∑τi)K, provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets. AU - Akopyan, Arseniy AU - Balitskiy, Alexey AU - Grigorev, Mikhail ID - 1064 IS - 4 JF - Discrete & Computational Geometry SN - 01795376 TI - On the circle covering theorem by A.W. Goodman and R.E. Goodman VL - 59 ER - TY - GEN AB - We prove that any convex body in the plane can be partitioned into m convex parts of equal areas and perimeters for any integer m≥2; this result was previously known for prime powers m=pk. We also give a higher-dimensional generalization. AU - Akopyan, Arseniy AU - Avvakumov, Sergey AU - Karasev, Roman ID - 75 TI - Convex fair partitions into arbitrary number of pieces ER - TY - JOUR AB - We introduce planar matchings on directed pseudo-line arrangements, which yield a planar set of pseudo-line segments such that only matching-partners are adjacent. By translating the planar matching problem into a corresponding stable roommates problem we show that such matchings always exist. Using our new framework, we establish, for the first time, a complete, rigorous definition of weighted straight skeletons, which are based on a so-called wavefront propagation process. We present a generalized and unified approach to treat structural changes in the wavefront that focuses on the restoration of weak planarity by finding planar matchings. AU - Biedl, Therese AU - Huber, Stefan AU - Palfrader, Peter ID - 481 IS - 3-4 JF - International Journal of Computational Geometry and Applications TI - Planar matchings for weighted straight skeletons VL - 26 ER -