Vladimir Boltyanski

Excursions into combinatorial geometry

The book deals with the combinatorial geometry of convex bodies in finite-dimensional spaces. A general introduction to geometric convexity is followed by the investigation of d-convexity and H-convexity, and by various applications. Recent research is discussed, for example the three (generally unsolved) problems from the combinatorial geometry of convex bodies: the Szoekefalvi-Nagy problem, the Borsuk problem, the Hadwiger covering problem. These and related questions are then applied to a new class of convex bodies which is a natural generalization of the class of zonoids: the class of belt bodies. Finally open research problems are discussed.

Combinatorial geometry of belt bodies

In this paper we consider a new class of convex bodies which was introduced in [11]. This is the class of belt bodies, and it is a natural generalization of the class of zonoids (see the surveys [18, 28, 24]). While the class of zonoids is not dense in the family of all centrally symmetric, convex bodies, the class of belt bodies is dense in the set of all convex bodies. Nevertheless, we shall extend solutions of combinatorial problems for zonoids (cf. [2, 12]) to the class of belt bodies. Therefore, we first introduce the set of belt bodies by using zonoids as starting point. (To make the paper self-contained, a few parts of the approach from [11] are given repeatedly.) Second, complete solutions of three well-known (and generally unsolved) problems from the combinatorial geometry of convex bodies are given for the class of belt bodies. The first of these, connected with the names of I. Gohberg and H. Hadwiger, is the problem of covering a convex body with smaller homothetic copies, or the equivalent illumination problem. The second is the Szökefalvi-Nagy problem, which asks for the determination of the convex bodies whose families of translates have a given Helly dimension. The third problem concerns special fixing systems, a notion which is due to L. Fejes Tóth. These solutions consist of improved and more general approaches to recently solved problems (as in the case of the Helly-dimensional classification of belt bodies) or new results (as those concerning minimal fixing systems, providing also an answer to a problem of B. Grünbaum which is not only restricted to belt bodies).

Carathéodory's Theorem and H-Convexity

In 1976, V. Boltyanski introduced the functional md for compact, convex bodies. With the help of this functional, some theorems of combinatorial geometry were derived. For example, the first author obtained a Helly-type theorem, later some particular cases of the Szokefalvi?Nagy problem were resolved. Further on, exact estimates for the cardinalities of primitive fixing and hindering systems of compact, convex bodies were established, etc. In this article, we discuss the connection of the classical Caratheodory Theorem with the functional md.

A new step in the solution of the Szökefalvi-Nagy problem

A description of vector systemsH⊂Rd that satisfy the condition mdH=2 is given. With the help of this description the main result is obtained. It consists in a listing of all compact, 2-Helly-dimensional, convex bodiesM⊂Rd. The listing is made in terms of polar bodiesM*.

Star-shaped sets in normed spaces

We prove a generalization of the Krasnosel’ski theorem on star-shaped sets. Usingd-segments inn-dimensional Minkowski spaces instead of usual segments, the notions “d-visibility” and “d-star-shapedness” are introduced. Our main aim is to give necessary and sufficient conditions ford-star-shapedness in finite-dimensional normed spaces.

Minkowski addition of H -convex sets and related Helly-type theorems

A natural generalization of the usual convexity notion is the notion of H-convexity. In the first part of the paper we will completely answer the question when the Minkowski sum of H-convex sets is itself an H-convex set. Based on this we are able to present, in the second part, results in the spirit of combinatorial geometry, namely some Helly-type theorems for H-convex sets.

The Szökefalvi-Nagy Problem

Let M ⊂ R n be a compact, convex body. We consider the family T(M) of all its translates (Fig. 111) and denote by him M the Helly dimension of T(M): him M = him T(M). This chapter is devoted to the Helly-dimensional classification of compact, convex bodies. In other words, we are going to consider the following problem: to give a geometrical description of the compact, convex bodies M ⊂ R n satisfying him M = r, where 1 ≤ r ≤ n. By reasons which will be mentioned in this section, this problem is said to be the Szokefalvi-Nagy problem.

Some research problems

Let d(x,y) be a metric in the n-dimensional vector space R n (without any connection to the metric induced by the norm and the linear operations in R n ). We say that the metric d is invariant with respect to translations if d(x + a, y + a) = d(x,y) for any a, x, y ∈ R n . Furthermore, we say that a metric d is normable if there exists a norm ∥ · ∥ in R n such that d(x,y) =∥ x − y ∥ for any x, y ∈ R n . Finally, we say that a metric d is bounded if the set B = { x ∈ R n : d(o, x) ≤ 1 { is bounded in R n . The problem is to describe a condition under which a metric d in R n is normable.

Minimum Convex Partitions of Polygonal Domains

In this chapter we study a certain type of problems on partition of planar polygonal domains into a minimum number of convex pieces. More exactly, for a given family F of non-oriented directions in the plane we determine the minimum number of convex pieces into which an arbitrary planar polygon can be partitioned by linear cuts in the directions from F. Based on this approach, we investigate the complexity status of various partition problems, such as partitions into rectangles, trapezoids, triangles, convex polygons.

Median problems in location science

In this chapter we are concerned with problems of the following type: Given a finite set of weighted points in Euclidean n-space, n ≥ 2, we are interested in the location of an additonal point such that the sum of weighted distances to the given points is minimal. Historically correct, this is the (generalized) Fermat-Torricelli problem, and in location science it is also called the Steiner- Weber problem and the 1-median problem, respectively. It is our aim to give a complete mathematical approach to this problem, to present several historical corrections and to study various natural generalizations of it. These generalizations refer to the distance measure under consideration (namely, by extending the problem to finite-dimensional normed spaces) and to the geometric configuration itself (replacement of the searched optimal point by a hyperplane, or by a flat of some other dimension). In all these cases, we first establish some necessary position criteria with the help of which then algorithmical approaches are obtained. So this chapter might be interesting from the viewpoint of classical and computational geometry, location science, and other applied disciplines (such as, e.g., robust statistics).