Perimeter Approximation of Convex Discs in the Hyperbolic Plane and on the Sphere

Eggleston (Approximation to plane convex curves. I. Dowker-type theorems. Proc. Lond. Math. Soc. 7, 351–377 (1957)) proved that in the Euclidean plane the best approximating convex n-gon to a convex disc K is always inscribed in K if we measure the distance by perimeter deviation. We prove that the analogue of Eggleston’s statement holds in the hyperbolic plane, and we give an example showing that it fails on the sphere.

A new enigmatic species of broad-nosed weevil endemic to Brazil and its phylogenetic placement within the tribe Naupactini (Coleoptera: Curculionidae: Entiminae)

A new species of Naupactini (Curculionidae: Entiminae) endemic to Brazil, Espírito Santo and Minas Gerais states, is herein described. It resembles the monotypic genus Hadropus Schoenherr in its general appearance, particularly in the shape of the elytra, and the color of the vestiture, but the results of a cladistics analysis herein conducted suggest that it belongs to the genus Stenocyphus Marshall. This genus ranges in São Paulo, Rio de Janeiro and Espírito Santo, mainly in the Atlantic forest of Brazil, and includes three other species. Stenocyphus costae sp. nov., distinguishes from the remaining species of Stenocyphus by the green iridescent scaly vestiture, having long stiff setae on the two pairs of elytral tubercles; the more slender rostrum; the shorter antennae; the convex disc of the pronotum; and the shorter and broader elytra. This paper includes a cladogram of the Naupactini genera showing the phylogenetic position of the new species, its complete description, photographs of male and female habitus, line drawings of genitalia of both sexes, and a key of the Stenocyphus species.

Variance estimates for random disc-polygons in smooth convex discs

In this paper we prove asymptotic upper bounds on the variance of the number of vertices and the missed area of inscribed random disc-polygons in smooth convex discs whose boundary is C+2. We also consider a circumscribed variant of this probability model in which the convex disc is approximated by the intersection of random circles.

On Random Disc Polygons in Smooth Convex Discs

In this paper we generalize some of the classical results of Rényi and Sulanke (1963), (1964) in the context of spindle convexity. A planar convex disc S is spindle convex if it is the intersection of congruent closed circular discs. The intersection of finitely many congruent closed circular discs is called a disc polygon. We prove asymptotic formulae for the expectation of the number of vertices, missed area, and perimeter difference of uniform random disc polygons contained in a sufficiently smooth spindle convex disc.

On Random Disc Polygons in Smooth Convex Discs

In this paper we generalize some of the classical results of Rényi and Sulanke (1963), (1964) in the context of spindle convexity. A planar convex discSis spindle convex if it is the intersection of congruent closed circular discs. The intersection of finitely many congruent closed circular discs is called a disc polygon. We prove asymptotic formulae for the expectation of the number of vertices, missed area, and perimeter difference of uniform random disc polygons contained in a sufficiently smooth spindle convex disc.