An (n; s) Davenport-Schinzel sequence, for positive integers n and s, is a sequence composed of n distinct symbols with the properties that no two adjacent elements are equal, and that it does not contain, as a (possibly non-contiguous) subsequence, any alternation a \Delta \Delta \Delta b \Delta \Delta \Delta a \Delta \Delta \Delta b \Delta \Delta \Delta of length s + 2 between two distinct symbols a and b. The close relationship between Davenport-Schinzel sequences and the combinatorial structure of lower envelopes of collections of functions make the sequences very attractive because a variety of geometric problems can be formulated in terms of lower envelopes. A near-linear bound on the maximum length of Davenport-Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.

Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a range-searching problem. A typical range-searching problem has the following form. Let S be a set of n points in R d , and let R be a family of subsets; elements of R are called ranges . We wish to preprocess S into a data structure so that for a query range R, the points in S " R can be reported or counted efficiently. Typical examples of ranges include rectangles, halfspaces, simplices, and balls. If we are only interested in answering a single query, it can be done in linear time, using linear space, by simply checking for each point p 2 S whether p lies in the query range.

This thesis establishes a connection between the Helly theorems, a collection of results from combinatorial geometry, and the class of problems whichwe call Generalized Linear Programming, or GLP, which can be solved by combinatorial linear programming algorithms like the simplex method. We use these results to explore the class GLP and show new applications to geometric optimization, and also to prove Helly theorems. In general, a GLP is a set...

INTRODUCTION A geometric transversal is an affine subspace of R d , such as a point, line, plane or hyperplane, which intersects every member of a family of convex sets. Eduard Helly's celebrated theorem gives conditions for the members of a family of convex sets to have a point in common, i.e., a point transversal. In Section 1 we highlight some of the more notable theorems related to Helly's Theorem and point transversals. Section 2 is devoted to geometric transversal theory. 4.1 HELLY-TYPE THEOREMS In 1913, Eduard Helly proved the following theorem: Theorem 1 (Helly's Theorem) Let A be a finite family of at least d + 1 convex sets in R d . If every d + 1 members of A have a point in common, then there is a point common to all the members of A. The theorem also holds for infinite families

Let S be an ordered set of disjoint unit spheres in R 3. We show that if every subset of at most six spheres from S admits a line transversal respecting the ordering, then the entire family has a line transversal. Without the order condition, we show that the existence of a line transversal for every subset of at most 11 spheres from S implies the existence of a line transversal for S. Categories and Subject Descriptors: F.2.2 [Nonnumerical

We completely describe the structure of the connected components of transversals to a collection of n line segments in R³. Generically, the set of transversal to four segments consist of zero or two lines. We catalog the non-generic cases and show that n >= 3 arbitrary line segments in R³ admit at most n connected components of line transversals, and that this bound can be achieved in certain configurations when the segments are coplanar, or they all lie on a hyperboloid of one sheet. This implies a tight upper bound of n on the number of geometric permutations of line segments in R³.

We prove that a suitably separated family of n compact convex sets in R d can be met by k-flat transversals in at most O(k) d 2 " 2 k+1 \Gamma 2 k '` n k + 1 "k(d\Gammak) or, for fixed k and d, O(n k(k+1)(d\Gammak) ) different order types. This is the first non-trivial upper bound for 1 ! k ! d \Gamma 1, and generalizes (asymptotically) the best upper bounds known for line transversals in R d , d ? 2. Introduction Let A be a family of n compact convex sets in R d . A line transversal of the family A is a line that intersects every member of A. If the sets in A are City College, City University of New York, New York, NY 10031, U.S.A. (jegcc@cunyvm.cuny.edu). Supported in part by NSF grant DMS93-22475, NSA grant MDA904-95-H-1012, and PSC-CUNY grant 665343. y Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, U.S.A. (pollack@geometry.nyu.edu). Supported in part by NSF grants DMS9400293, CCR94-02640, and CCR94-24398. z Ohio Stat...

... "), for any " ? 0. This result has several applications: (i) a near-quadratic upper bound on the complexity of the region in 3- space enclosed between the lower envelope of one such collection of functions and the upper envelope of another collection; (ii) an efficient and simple divide-and-conquer algorithm for constructing lower envelopes in three dimensions; and (iii) a near-quadratic upper bound on the complexity of the space of all plane transversals of a collection of simply-shaped convex sets in three dimensions.

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Let A be a family of convex sets in R d . A line transversal to A is a line which intersects every member of A. More generally, a k-tranversal to A is an ane subspace of dimension k which intersects every member of A.