On spherically convex sets and Q-matrices

Let K be a closed spherically convex subset of S^n?1 that is contained in a hemisphere, and x?(K) the radial projection onto S^n?1 of the centroid of K. Then p^Tx?(K)>0 for all p ? K. A specialization of this result to spherical simplices is used to derive a necessary condition for Q-matrices, i.e., matrices for which every corresponding linear complementarity problem has at least one solution.     

Intersections ofk-element sets

LetF be a collection ofk-element sets with the property that the intersection of no two should be included in a third. We show that such a collection of maximum size satisfies .2715k+o(k)≦≦log₂ |F|≦.7549k+o(k) settling a question raised by Erdős. The lower bound is probabilistic, the upper bound is deduced via an entropy argument. Some open questions are posed. This research has been supported in part by the Office of Naval Research under Contract N00014-76-C-0366. Supported in part by a NSF postdoctoral Fellowship.     

Intersections of maximal staircase sets

A compact set is staircase connected if every two points a, b ∈ S can be connected by an x-monotone and y-monotone polygonal path with sides parallel to the coordinate axes. In [5] we have introduced the concepts of staircase k-stars and kernels. In this paper we prove that if the staircase k-kernel is not empty, then it can be expressed as the intersection of a covering family of maximal subsets of staircase diameter k of S.      

Intersections of Markov random sets

Probability Theory and Related Fields -     

Almost convex sets

The concept of a wedgoid, a generalized wedge, is introduced and studied. Almost all convex sets are characterized by the fact that balls can be separated from such a set by means of wedgoids.     

Protecting convex sets

A point-setS is protecting a collection F =T ₁,T ₂,..., _n ofn mutually disjoint compact sets if each one of the setsT _i is visible from at least one point inS; thus, for every setT _i ∈F there are points xS andy T _i such that the line segment joining x to y does not intersect any element inF other thanT _i . In this paper we prove that [2(n-2)/3] points are always sufficient and occasionally necessary to protect any family F ofn mutually disjoint compact convex sets. For an isothetic family F, consisting ofn mutually disjoint rectangles, [n/2] points are always sufficient and [n/2] points are sometimes necessary to protect it. IfF is a family of triangles, [4n/7] points are always sufficient. To protect families ofn homothetic triangles, [n/2] points are always sufficient and [n/2] points are sometimes necessary.     

Extremal convex sets

Letf be an extended real valued function on the classK ⁿ of closed convex subsets of euclideann-dimensional space. A setK∈K ⁿ is said to bef-maximal if the conditionsK′∈K ⁿ ,K?K′,K≠K′ implyf(K)<f(K′), andf-minimal ifK′∈K ⁿ,K′∈K,K′≠K impliesf(K′)<f(K). In the cases whenf is the circumradius or inradius allf-maximal andf-minimal sets are determined. Under a certain regularity assumption a corresponding result is obtained for the minimal width. Moreover, a general existence theorem is established and a result concerning the existence of extremal sets with respect to packing and covering densities is proved.     

Nine convex sets determine a pentagon with convex sets as vertices

It is proved that if ℱ is a family of nine pairwise disjoint compact convex sets in the plane such that no member of ℱ is contained in the convex hull of the union of two other sets of ℱ, then ℱ has a subfamily ℱ′ with five elements such that no member of ℱ′ is contained in the convex hull of the union of the other sets of ℱ′.     

Intersections and limits of regenerative sets

Summary Regenerative subsets of ℝ constitute an analog of classical renewal processes. Limits and intersections of independent regenerative sets are discussed. These ideas are related to the usual quantities associated with subordinators. Portions of this work appeared in the author's Ph.D. dissertation, written under the direction of Kenneth J. Hochberg of Case Western Reserve University Research partially supported by National Science Foundation Grant MCS83-01080     

Empty convex polygons in almost convex sets

A finite set of points, in general position in the plane, is almost convex if every triple determines a triangle with at most one point in its interior. For every ℓ ≥ 3, we determine the maximum size of an almost convex set that does not contain the vertex set of an empty convex ℓ-gon. Partially supported by grants T043631 and NK67867 of the Hungarian NFSR (OTKA).     

Bayes estimation with spherically symmetric,convex loss

Summary It is desired to estimate a parameter with the loss function of the formL(θ, a)=W(‖θ−a‖), where is convex, differentiable, and non-decreasing. With this structure a characterization of Bayes estimators is given. Also it is noted that if the sample space, , for the observation,X, is a complete separable metric space then a Bayes estimator exists.     

Asymptotics of the convex hull of spherically symmetric samples

In this paper we consider the convex hull of a spherically symmetric sample in R^d. Our main contributions are some new asymptotic results for the expectation of the number of vertices, number of facets, area and the volume of the convex hull assuming that the marginal distributions are in the Gumbel max-domain of attraction. Further, we briefly discuss two other models assuming that the marginal distributions are regularly varying or O-regularly varying.