It is shown that there exists a number d 0 < 8(2 p 3?3)=3 = 1:237604::, such that every compact convex set K with an interior point admits a covering of the plane with density smaller than or equal to d 0. This improves on the previous result 9], which showed that a density of 8(2 p 3 ? 3)=3 can always be obtained. Since the thinnest covering of the plane with congruent circles is of density 2=...

We show that the number of unit-area triangles determined by a set S of n points in the plane is O(n), improving the earlier bound O(n) of Apfelbaum and Sharir [2]. We also consider two special cases of this problem: (i) We show, using a somewhat subtle construction, that if S consists of points on three lines, the number of unit-area triangles that S spans, with one vertex on each of the lines...

On a bounded strongly pseudo-convex domain X in C with a Lipschitz boundary, we prove that the ∂̄−Neumann operator N can be extended as a bounded operator from Sobolev (−1/2)−spaces to the Sobolev (1/2)−spaces. In particular, N is compact operator on Sobolev (−1/2)−spaces.

We consider the problem of guarding curvilinear art galleries. A closed arc a joining two points, p and q, in the plane is called a convex arc if the curve obtained by joining a with the line segment pq encloses a convex set. A piece-wise convex polygon P with vertices v0, . . . , vn−1 is the region bounded by a set {a0, . . . , an−1} of n convex arcs with pairwise disjoint interiors such that ...

In the present paper we establish an abstract principle of condensation of singularities for families consisting of set-valued mappings. By using it as a basic tool, the condensation of the singularities and the equicontinuity of certain families of generalized convex set-valued mappings are studied. In particular, a principle of condensation of the singularities of families of closed convex pr...

1. In the theory of (F)-spaces,1 the following two questions arise: (A) Let £ be a metrizable locally convex space; is it true that if every bounded set in E is separable, the space E is separable? (B) Let £ be a metrizable locally convex space, E its completion; is it true that every bounded set in E is contained in the closure of a bounded set of El The purpose of this note is to prove that, ...

Let us start with the basic model of Search Theory. An n-element set S is given, one of its elements, say x, is distinguished, the goal is to find x. Questions of type ”x ∈ A?” can be asked, where A is a subset of S. The unknown x should be determined on the base of the answers to these questions. In general one cannot use every subset A. A family A ⊂ 2 is given, the question sets A can be chos...

Given a universe (a set) U and a property P, (closed under inclusions, for subsets of U). Results of the type “if every subset of cardinality μ of a finite family F ⊂ U has property P, then the entire family F has property P”are called Helly type theorems. The minimum number μ for which the result is true is called the Helly number of the Helly-type theorem (U ,P , μ). In the case of Helly’s cl...

A line meeting a family of pairwise disjoint convex sets induces two permutations of the sets. This pair of permutations is called a geometric permutation. We characterize the possible triples of geometric permutations for a family of disjoint translates in the plane. Together with earlier studies of geometric permutations this provides a complete characterization of realizable geometric permut...