Ted Lewis

Geometric permutations for convex sets

Abstract A geometric permutation is the pair of permutations formed by a common transversal for a finite family of disjoint convex sets in the plane. It is shown that if A is a family of n convex and pairwise disjoint segments in the plane then the number of geometric permutations formed by all common transversals of A cannot be more than n .

The different ways of stabbing disjoint convex sets

We construct a family ofn disjoint convex set in ℝd having (n/(d−1))d−1 geometric permutations. As well, we complete the enumeration problem for geometric permutations of families of disjoint translates of a convex set in the plane, settle the case for cubes in ℝd, and construct a family ofd+1 translates in ℝd admitting (d+1)!/2 geometric permutations.

Geometric permutations and common transversals

The object of this paper is to study how many essentially different common transversals a family of convex sets on the plane can have. In particular, we consider the case where the family consists of pairwise disjoint translates of a single convex set.

Geometric permutations of disjoint translates of convex sets

Abstract The object of this paper is to study how many essentially different common transversals a family of convex sets on the plane can have. In particular we consider the case where the family consists of pairwise disjoint translates of a single convex set.

Cutting families of convex sets

A family of convex sets in the plane admits a common transversal if there is a straight line which intersects (cuts) each member of the family. It is shown that there is a positive integer k such that for any compact convex set C in the plane and for any finite family & of pairwise disjoint translates of C: If each 3-membered subfamily of & admits a common transversal then there is a subfamily

Common supports as fixed points

of & such that i& admits a common transversal and \â \ 9> \ <,k.

Cutting rectangles in the plane

A family S of sets in Rd is sundered if for each way of choosing a point from r≤d+1 members of S, the chosen points form the vertex-set of an (r−1)-simplex. Bisztriczky proved that for each sundered family S of d convex bodies in Rd, and for each partition (S′, S″), of S, there are exactly two hyperplanes each of which supports all the members of S and separates the members of S′ from the members of S″. This note provides an alternate proof by obtaining each of the desired supports as (in effect) a fixed point of a continuous self-mapping of the cartesian product of the bodies.

On the recognition of S-systems

Let F be a planar family of pairwise disjoint translates of a parallelogram. It is shown that if every four members of F are intersected by some straight line, then there is a straight line which intersects all but at most two members of F. If it is instead assumed that every two members of F are intersected by some line that is parallel to one of the edges of the parallelogram, then the same conclusion holds.

Two counterexamples concerning transversals for convex subsets of the plane

Abstract A sequence (A1,…,An) of nonempty subsets of R m is an S-system if for each choice of a1∈A1,…,an∈An, the nullspace of the matrix [a1,…,an] is a line that penetrates the strictly positive orthant of R n. For the case in which each Aj is a sign cone, S-systems are basic to the study of the sign-solvability of systems of linear equations; in that case, they can be recognized in time O(m2). The present paper provides several characterizations of general S-systems and describes a polynomial-time algorithm (based on linear programming) for the recognition of such systems when the Aj are arbitrary finitely presented convex cones.