Csaba Vincze (2013)

University of Debrecen

''The study of convex sets is a branch of geometry, analysis and linear algebra that has numerous connections with other areas of mathematics and serves to unify many apperantly diverse mathematical phenomena'' (Victor Klee).

The systematic study of convex sets is a relatively young theory. The first monograph [10] was published by Bonnesen and Frenchel in 1934. In the middle of the 20th century lots of useful applications of convex sets were discovered. According to the importance of these applications convexity is a prosperous subject up to this day. In what follows we collectsome basic facts from the theory in different levels. The area of the material is the classical Euclidean space of dimension n. It is broad enough to include many of important applications. On the other hand this setting allows us to simplify many of the proofs.

The first chapter is divided into six sections 1.1-1.6. From 1.1 to 1.3 we summarize the elements of the linear algebra, topology, affine and convex sets to prepare the classical theorems of convex geometry. These are discussed in chapter 2 (Carathéodory's theorem) and chapter 3 (Radon's lemma and Helly's theorem). Refinements and generalizations such as the colorful Carathéodory's theorem due to I. Bárány [4] are also considered. The problem of separating and supporting hyperplanes can be found in chapter 7 to prepare the classical structure theorem of convex polyhedra (chapter 9) in the space of dimensionthree (vertices, edges, facets). The introductory level can be represented by the following diagram:

$\begin{array}{lllll}sections\text{}\mathrm{1.1,1.2}\text{}\text{}and\text{}\text{}1.3\text{}& \text{}\to \text{}& chapter\text{}2\text{}\to \text{}& chapter\text{}3\text{}& \\ \mathbf{}\downarrow & & & & \\ chapter\text{}7\text{}& \text{}\to \text{}& \text{}chapter\text{}9.& & \end{array}$ |

The mathematical prerequisites for the study of this level are linear algebra and basic point-set topology. Generalizations and applications can be found in chapters 4, 5, 6 and 8. Applications in the art gallery geometry are presented by Krasnosselsky's theorem (chapter 5) [37]. To illustrate that the study of convex sets has numerous connections with other areas of mathematics we tend to present some surprising applications such as affine [44] and convex [3] separations between functions (section 4.3). As a recent trend of the research we also refer to the problem of separation by members of a given linear interpolation family [46]. The common tool of these applications is the classical Helly's theorem and the proofs reflect the geometric feature of the problem. Kirchberger's theorem (chapter 8) [37] has a nice application in the approximation theory: how to find the best affine approximation for a given finite set of points. Another intensively studied area of the research is the generalization of the classical results for star-shaped sets (chapter 6). Among others the literature contains a star-shaped version of Krein-Milman's theorem [40] and Helly type theorems for intersections of star-shaped sets [15], [16] and [9]. Although Minkowski geometry is a very natural attached theory to convex sets it is only partially discussed in chapter 7 (section 7.3). It also appears in some applications (section 10.2). For lack of space another important parts of the theory are also missing. For example the study of inequalities concerning volumes of compact convex sets appears only as a subsection 4.2.1 or in connection with X-ray functions (section 10.3). Such kind of illustrative materials present new starting points for those interested in modern aspects of geometry [7]. Nowadays they are self-supporting branches of geometry together with basic monographs such as [53], [26] and [52]. The advanced course can be represented by the following diagram:

$\begin{array}{llllll}sections\text{}\mathrm{1.1,1.2}\text{}\text{}and\text{}\text{}1.3\text{}& \text{}\to \text{}& \text{}chapter\text{}2\text{}\to \text{}& chapter\text{}3\text{}& & \\ \downarrow & & & \downarrow & & \\ chapter\text{}7\text{}& \text{}\to \text{}& chapter\text{}9\text{}& chapter\text{}4\text{}& \text{}\to \text{}& chapter\text{}6\text{}\\ \downarrow & & & \downarrow & & \\ chapter\text{}8\text{}& & & chapter\text{}5.\text{}& & \end{array}$ |

To take more steps forward we need some basic facts about metric properties of the space of convex compact sets and the foundations of the theory of convex functions (the natural domains for these functions are convex sets). We summarize them in the first chapter from 1.4 to 1.6. Chapters 10 and 11 present some special topics related to the convexity: Erdős-Vincze's theorem [24] and the theory of generalized conics.

$\begin{array}{lllll}chapter\text{}10\text{}& \text{}\leftarrow \text{}& chapter\text{}1\text{}& \text{}\to \text{}& chapter\text{}11\text{}\\ & & \downarrow & & \\ & & chapter\text{}12.\text{}& & \\ & & & & \end{array}$ |

The object of the generalized conics' theory (chapter 10) is the investigation of subsets in the space all of whose points have the same average distance from the set of foci. The "average" can be realized in several ways from classical (discrete) means to integration over the set of foci. In a significant part of typical situations the common feature of functions measuring the average distance is the convexity. They also satisfy a kind of growth condition. These properties imply that the (lower) level sets are compact convex subsets in the space bounded by compact convex hypersurfaces. They are called generalized conics. The most important discrete cases are polyellipses with the classical arithmetic mean to calculate the average Euclidean distance from the elements of a finite point-set and lemniscates (with the classical geometric mean to calculate the average Euclidean distance from the elements of a finite point-set). Lemniscates in the plane play a central role in the theory of approximation in the sense that polynomial approximations of holomorphic functions can be interpreted as approximations of curves with lemniscates. In terms of algebra we speak about the roots of polynomials (in terms of geometry we speak about the focuses of lemniscates). Endre Vázsonyi posed the problem whether the polyellipses (as the additive version of lemniscates) have the same approximating property by increasing the number of the foci or not. The answer is negative as a theorem due to P. Erdős and I. Vincze states. The proof can be found in chapter 11. In the literature we can find many generalizations of conics [47] and [30]. Computational difficulties are also significant in the theory. For the case of polyellipses see [23]. To computethe integral of the Euclidean distance along a curve to a given point is impossible in general. In case of a circle in the space we immediately have elliptic integrals. Nevertheless the best (recent) results [2] and [49] on elliptic integrals and Gaussian hypergeometric function allow us to develop a kind of theory of circular (generalized) conics [56], see also [57]. This is a partial motivation why to substitute the Euclidean distance with a more computable way of measuring the distance between the points in the space. Interesting applications in geometric tomography were found by measuring the average taxicab distance of points to a given subset. This is closely related to the coordinate X-ray functions (up to a multiplicative constant) which are typical sources of information about unknown bodies [26]. Beyond the (parallel or point) X-rays, projections and sections of sets we can also refer to the so-called angle function. The notion was introduced by J. Kincses [33] together with the problem of determination.

The last chapter is devoted to Radström's embedding theorem [48]. The theorem states that the collection of convex compact sets can be considered as a (convex) cone in an infinitely dimensional normed space, see also [17]. It is a natural idea to apply the calculus to volume, X-rays, anglefunctions etc. as mappings defined on the cone of convex compact sets. In many important particular cases we have nice properties. For example the Brunn-Minkowski inequality (for the concavity of the nth root of the n-dimensional Lebesgue measure) implies that the volume belongs to the class of quasi-concave functions having convex (upper) level sets. Another example (for coordinate X-rays) can be found in [58]. The last chapter of this material presents a new starting point of the investigation too.