A random polygon is the convex hull of uniformly distributed random points in a convex body K ⊂ R. General upper bounds are established for the variance of the area of a random polygon and also for the variance of its number of vertices. The upper bounds have the same order of magnitude as the known lower bounds on variance for these functionals. The results imply a strong law of large numbers ...

Various classes of finite-dimensional closed convex cones are studied. Equivalent characterizations of pointed cones, pyramids and rational pyramids are given. Special class of regular cones, corresponding to "continuous linear" quasiorderings of integer vectors is introduced and equivalently characterized. It comprehends both pointed cones and rational pyramids. Two different ways of determini...

CONFIGURATIONS IN ALGEBRAIC GEOMETRY

Human adults from diverse cultures share intuitions about the points, lines, and figures of Euclidean geometry. Do children develop these intuitions by drawing on phylogenetically ancient and developmentally precocious geometric representations that guide their navigation and their analysis of object shape? In what way might these early-arising representations support later-developing Euclidean...

Let R and B be disjoint point sets such that R ∪ B is in general position. We say that B encloses by R if there is a simple polygon P with vertex set B such that all the elements in R belong to the interior of P . In this paper we prove that if the vertices of the convex hull of R ∪ B belong to B, and |R| ≤ |Conv(B)| − 1 then B encloses R. The bound is tight. This improves on results of a previ...