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First, we give a proof of the long-standing Kneser–Poulsen conjecture in the Euclidean plane. Although that result is 2-dimensional its proof is higher dimensional. Second, we prove an analogue of the Kneser–Poulsen conjecture for hemispheres in spherical d-space. Third, we give a proof of a Kneser–Poulsen-type theorem for convex polyhedra in hyperbolic 3-space.

In this chapter we study the structure of approximate solutions of autonomous variational problems with convex integrands. The main goal is to study the structure of approximate solutions of the problems in the regions close to the endpoints of the domains. The results of this chapter provide the full description of the structure of solutions of the convex autonomous problems.

Measures on flag manifolds have been recently used to describe local properties of convex bodies and more general sets in \({\mathbb{R}}^{d}\). Here, we provide a systematic account of flag measures for convex bodies, we collect various properties of flag measures and we prove some new results. In particular, we discuss mixed flag measures for several bodies and we present formulas for (mixed) flag measures of generalized zonoids.

In this chapter we study nonoccurrence of the Lavrentiev phenomenon for a large class of nonconvex nonautonomous constrained variational problems. A state variable belongs to a convex subset H of a Banach space X with nonempty interior. Integrands belong to a complete metric space of functions \(\mathcal{M}_{B}\) which satisfy a growth condition common in the literature and are Lipschitzian on bounded sets. This space will be described below. In [97] we considered a class of nonconstrained variational problems with integrands belonging to a subset \(\mathcal{L}_{B} \subset \mathcal{M}_{B}\) and showed that for \(f \in \mathcal{L}_{B}\) the following property holds

In this chapter, we introduce an extension of the theory of convex polyhedral sets (resp., convex bodies) to the family of intersections of finitely many congruent balls, called ball-polyhedra (resp., to the family of intersections of not necessarily finitely many congruent balls, called spindle convex bodies). The basic properties of ball-polyhedra (resp., spindle convex bodies) that are discussed here include separation and support properties for spindle convex bodies, a Caratheodory-type theorem for spindle convex hulls, and an Euler–Poincare-type formula for ball-polyhedra in d-dimensional Euclidean space. In addition, we discuss (generalized) billiard trajectories in disk-polygons and an analogue of Blaschke–Lebesgue theorem for disk-polygons. Furthermore, we investigate the problem of characterizing the edge-graphs of ball-polyhedra in Euclidean 3-space. Another topic, discussed in more details, is on global and local rigidity of ball-polyhedra in Euclidean 3-space. Finally, we investigate the status of the long-standing illumination conjecture of V. G. Boltyanski and H. Hadwiger from 1960 for ball-polyhedra (resp., spindle convex bodies) in Euclidean d-space.

In this paper it will be demonstrated how functional analytic tools from duality theory can be used to give interesting characterizations of stochastic order relations for discrete distributions in terms of mass transfer principles. A general result for a large class of integral stochastic orders will be derived, and it will be shown that this applies to many important examples like usual stochastic order, convex order, supermodular order, directional convex order, and orthant orders.

Considering the fact that polyhedra have been studied for so long, it is rather surprising that there has been no exhaustive study of their history. But we are very lucky that the authors of four modern classics on the theory of polyhedra—Bruckner, Coxeter, Fejes-Toth and Grunbaum—were interested in historical information and provided detailed historical notes in their books. I propose to present an outline of the milestones in the history of the subject, putting together the thread of what happened as the theory developed. I will pay special attention to regularity concepts.

This chapter includes six “recipes” for making polyhedra, devised by famous polyhedra31.3pc]Please provide affiliation for “Arthur Vi Hart”. chefs. Some recipes are for beginners, others are intermediate or advanced. You can use these recipes, or devise your own. Building models is fun, and will give you a deeper understanding of the chapters that follow.

Plane pictures of three-dimensional convex polyhedra, plane sections of three-dimensional Dirichlet tessellations, and flat spider webs with tension in all the threads are essentially the same geometric object. At the root of this remarkable coincidence is a single geometric diagram that permits us to offer a unified image of the connections among these and other objects. Some hints of these connections are more than a century old, but others are very recent. We begin with an historical sketch.

What is a polyhedron? The question is short, the answer is long. Although you may never have heard of the Polyhedron Kingdom before, it is nearly as vast and as varied as the animal, mineral, and vegetable kingdoms (and it overlaps all three of them). There are aristocrats and workers, families and individuals, old polyhedra with long and interesting histories and young polyhedra born yesterday or the day before. In this kingdom you can take a walking tour of polyhedral architecture, visit a nature preserve and an art gallery and an artisans’ polyhedra fair. As you stroll along you may even glimpse polyhedral ghosts from four-dimensional space.

In this chapter we investigate polyhedra in Euclidean 3-space, E 3, without self-intersections and with some local and global properties related to those of the Platonic solids. A polyhedron is the geometric realization of a compact 2-manifold in E 3 such that its 2-faces are (not necessarily convex) plane polygons bounded by finitely many line segments. Adjacent faces and edges are not coplanar. A flag of a polyhedron P is any triple consisting of a vertex, an edge, and a face of P, all mutually incident.

A graph G is called generically minimally rigid in ℝ d if, for any choice of sufficiently generic edge lengths, it can be embedded in ℝ d in a finite number of distinct ways, modulo rigid transformations. Here, we deal with the problem of determining tight bounds on the number of such embeddings, as a function of the number of vertices. The study of rigid graphs is motivated by numerous applications, mostly in robotics, bioinformatics, sensor networks, and architecture. We capture embeddability by polynomial systems with suitable structure so that their mixed volume, which bounds the number of common roots, yields interesting upper bounds on the number of embeddings. We explore different polynomial formulations so as to reduce the corresponding mixed volume, namely by introducing new variables that remove certain spurious roots and by applying the theory of distance geometry. We focus on \({\mathbb{R}}^{2}\) and \({\mathbb{R}}^{3}\), where Laman graphs and 1-skeleta (or edge graphs) of convex simplicial polyhedra, respectively, admit inductive Henneberg constructions. Our implementation yields upper bounds for \(n \leq 10\) in \({\mathbb{R}}^{2}\) and \({\mathbb{R}}^{3}\), which reduce the existing gaps and lead to tight bounds for \(n \leq 7\) in both \({\mathbb{R}}^{2}\) and \({\mathbb{R}}^{3}\); in particular, we describe the recent settlement of the case of Laman graphs with seven vertices. Our approach also yields a new upper bound for Laman graphs with eight vertices, which is conjectured to be tight. We also establish the first lower bound in \({\mathbb{R}}^{3}\) of about 2. 52 n , where n denotes the number of vertices.

Let \(t(\mathcal{C})\) be the number of tangent pairs among a set \(\mathcal{C}\) of n Jordan regions in the plane. Pach et al. [Tangencies between families of disjoint regions in the plane, in Proceedings of the 26th ACM Symposium on Computational Geometry (SoCG 2010), Snowbird, June 2010, pp. 423–428] showed that if \(\mathcal{C}\) consists of convex bodies and its intersection graph is bipartite, then \(t(\mathcal{C}) \leq 4n - \Theta (1)\), and, moreover, there are such sets that admit at least \(3n - \Theta (\sqrt{n})\) tangencies. We close this gap and generalize their result by proving that the correct bound is \(3n - \Theta (1)\), even when the intersection graph of \(\mathcal{C}\) is only assumed to be triangle-free.

In a convex n-gon, let \({d}_{1} > {d}_{2} > \cdots \) denote the set of all distances between pairs of vertices, and let m i be the number of pairs of vertices at distance d i from one another. Erdős, Lovasz, and Vesztergombi conjectured that \(\sum\nolimits_{i\leq k}{m}_{i} \leq kn\). Using a new computational approach, we prove their conjecture when k ≤ 4 and n is large; we also make some progress for arbitrary k by proving that \(\sum\nolimits_{i\leq k}{m}_{i} \leq (2k - 1)n\). Our main approach revolves around a few known facts about distances, together with a computer program that searches all distance configurations of two disjoint convex hull intervals up to some finite size. We thereby obtain other new bounds, such as m 3 ≤ 3n∕2 for large n.

Let P be a set of points in the plane in general position. Any three points \(x,y,z \in P\) determine a triangle \(\Delta (x,y,z)\) of the plane. We say that \(\Delta (x,y,z)\) is empty if its interior contains no element of P. In this chapter, we study the following problems: What is the size of the largest family of edge-disjoint triangles of a point set? How many triangulations of P are needed to cover all the empty triangles of P? We also study the following problem: What is the largest number of edge-disjoint triangles of P containing a point q of the plane in their interior? We establish upper and lower bounds for these problems.

In this chapter, we present a complete characterization of the smallest sets that block all the simple spanning trees (SSTs) in a complete geometric graph. We also show that if a subgraph is a blocker for all SSTs of diameter at most 4, then it must block all simple spanning subgraphs and, in particular, all SSTs. For convex geometric graphs, we obtain an even stronger result: Being a blocker for all SSTs of diameter at most 3 is already sufficient for blocking all simple spanning subgraphs.

This chapter provides a concrete introduction to several advanced topics in optimization theory. Specifying an interior feasible point is the first issue that must be faced in applying a barrier method. Given an exterior point, Dykstra’s algorithm [21, 70, 79] finds the closest point in the intersection \(\cap _{i=0}^{r-1}C_{i}\) of a finite number of closed convex sets. If C i is defined by the convex constraint \(h_{i}(\boldsymbol{x}) \leq 0\), then one obvious tactic for finding an interior point is to replace C i by the set \(C_{i}(\epsilon ) =\{ \boldsymbol{x} : h_{j}(\boldsymbol{x}) \leq -\epsilon \}\) for some small e > 0. Projecting onto the intersection of the C i (e) then produces an interior point.

Ellipses and ellipsoids form a well-established special class of convex surfaces, primarily due to a wide range of their applications in various mathematical disciplines. The present survey deals with a natural extension of this class to that of convex quadrics. It contains a classification of convex quadrics of the Euclidean space R n and describes, in terms of plane quadric sections, their various characteristic properties among all convex hypersurfaces of R n , possibly unbounded.

There are several instances (e.g., [2, p. 255], [3, p. 255]) in the general equilibrium theory of economics where one is interested in the following question: Given sets {X i } i = 1 n in a vector space V, how close is the vector sum $${\sum }_{i=1}^{n}{X}_{ i} = \left \{{\sum }_{i=1}^{n}{x}_{ i} : {x}_{i} \in {X}_{i}\right \}$$ (4.1) to being convex?

In this chapter, we shall present existence results for some classes of initial value problems for partial hyperbolic differential inclusions with fractional order involving the Caputo fractional derivative, when the right-hand side is convex as well as nonconvex valued. Some results rely on the nonlinear alternative of Leray–Schauder type. In other results, we shall use the fixed-point theorem for contraction multivalued maps due to Covitz and Nadler.

We investigate the optimized configuration of sensor cameras to be placed in a suitably defined three-dimensional cubic region E, in order to maximize the coverage of a completely embedded cube C⊂E. The sensing regions associated with each of the cameras are convex, but not necessarily identical. In order to handle this important practical problem, we present mixed integer linear programming (MILP) and mixed integer nonlinear programming (MINLP) model formulations and propose corresponding solution approaches. Illustrative numerical results are presented, and certain application aspects are also discussed.

In Chapter 4, we showed that when a graph is embedded in a suitably constructed Markov decision process, the associated convex domain of discounted occupational measures is a polyhedron with extreme points corresponding to all spanning subgraphs of the given graph. Furthermore, from Theorem 4.1 we learned that a simple cut of the above domain yields a polyhedron the extreme points of which correspond to only two possible types: Hamiltonian cycles and convex combinations of short and noose cycles. These properties, naturally, suggest certain algorithmic approaches to searching for Hamiltonian cycles.

We have seen in Chap. 4 that necessary conditions for optimality follow from separation results using convex approximations for the reachable set from a point. If the reachable sets are known exactly, not only necessary conditions, but complete solutions can be obtained for related optimal control problems (e.g., the time-optimal control problem). In general, determining these sets is as difficult a problem as solving an optimal control problem.

Newton’s aerodynamic problem consists in minimizing the resistance to the translational motion of a three-dimensional body moving in a homogeneous medium of resting particles. The particles do not interact with each other and reflect off elastically in collisions with the body. This problem has been considered for various classes of admissible bodies. In Newton’s initial setting[40], the class of admissible bodies consisted of convex axisymmetric bodies of fixed length and width, that is, bodies inscribed in a fixed right circular cylinder. The problem was later considered for various classes of (convex and axisymmetric) bodies, for example, for bodies whose front generator has a fixed length (and whose width is also fixed)[32,4], for bodies of fixed volume[5], and so on. A major step forward was made in the 1990s, when unexpected and striking results were obtained for some classes of nonaxisymmetric bodies and, later, for nonconvex bodies[8,11,12,16,17,18,29,30,31]). However, the authors kept the initial assumption that the body must have a fixed length and width, that is, can be inscribed in a fixed right circular cylinder.

We provide L p →L q estimates for a class of Fourier multipliers supported in convex cones of { R } n+1. In particular, we consider cones whose boundary has n−1 nonvanishing principal curvatures and cones which are the convex envelope of N linearly independent half lines passing through the origin of { R } n+1. In some case our estimates are best possible.

The class of mathematical programs with complementarity constraints (MPCCs) constitutes a powerful modeling paradigm. In an effort to find a global optimum, it is often useful to examine the relaxation obtained by omitting the complementarity constraints. We discuss various methods to tighten the relaxation by exploiting complementarity, with the aim of constructing better approximations to the convex hull of the set of feasible solutions to the MPCC, and hence better lower bounds on the optimal value of the MPCC. Better lower bounds can be useful in branching schemes to find a globally optimal solution. Different types of linear constraints are constructed, including cuts based on bounds on the variables and various types of disjunctive cuts. Novel convex quadratic constraints are introduced, with a derivation that is particularly useful when the number of design variables is not too large. A lifting process is specialized to MPCCs. Semidefinite programming constraints are also discussed. All these constraints are typically applicable to any convex program with complementarity constraints. Computational results for linear programs with complementarity constraints (LPCCs) are included, comparing the benefit of the various constraints on the value of the relaxation, and showing that the constraints can dramatically speed up the solution of the LPCC.

In this paper we survey recent work on the perspective reformulation approach that generates tight, tractable relaxations for convex mixed integer nonlin- ear programs (MINLP)s. This preprocessing technique is applicable to cases where the MINLP contains binary indicator variables that force continuous decision variables to take the value 0, or to belong to a convex set. We derive from first principles the perspective reformulation, and we discuss a variety of practical MINLPs whose relaxation can be strengthened via the perspective reformulation. The survey concludes with comments and computations comparing various algorithmic techniques for solving perspective reformulations.

The study of N-body systems and their simulation with various models always excited the scientific interest. Here we present an N-body model that has been under investigation in the last 10 years and is called the ring problem of (N+1) bodies, or otherwise the regular polygon problem of (N+1) bodies. In what follows, we give an overview of the scientific work that has been done through all these years, as well as the major results obtained so far.

In this chapter, we provide a non-exhaustive account of decomposition algorithms for solving structured large scale convex and non-convex optimization problemswith major emphasis on several splitting approaches based on the classical or modified augmented Lagrangian functions. This study covers last 40 years of research on theoretical properties of augmented Lagrangians.

Construction of unimodal random probability measures on finite dimensional Euclidean space is considered. The approach based on Bayesian nonparametric models and Convexity Theory. Specifically, the proposed model makes use of the special properties of convex sets and Choquet’s theorem. As a result, we get random probability measures that admit derivatives almost everywhere in R d .

The notion of a convex cone, which lies between that of a linear subspace and that of a convex set, is the main topic of this chapter. It has been very fruitful in many branches of nonlinear analysis. For instance, closed convex cones provide decompositions analogous to the well-known orthogonal decomposition based on closed linear subspaces. They also arise naturally in convex analysis in the local study of a convex set via the tangent cone and the normal cone operators, and they are central in the analysis of various extensions of the notion of an interior that will be required in later chapters.

Convex optimization is one of the main areas of application of convex analysis. This chapter deals with the issues of existence and uniqueness in minimization problems, and investigates properties of minimizing sequences.

In this chapter, we deepen our study of (maximally) monotone operators. The main results are Minty’s theorem, which conveniently characterizes maximal monotonicity, and the Debrunner–Flor theorem, which concerns the existence of a maximally monotone extension with a prescribed domain localization. Another highlight is the fact that the closures of the range and of the domain of a maximally monotone operator are convex, which yields the classical Bunt–Motzkin result concerning the convexity of Chebyshev sets in Euclidean spaces. Results on local boundedness, surjectivity, and single-valuedness are also presented.

The Douglas–Rachford iteration scheme, introduced half a century ago in connection with nonlinear heat flow problems, aims to find a point common to two or more closed constraint sets. Convergence of the scheme is ensured when the sets are convex subsets of a Hilbert space, however, despite the absence of satisfactory theoretical justification, the scheme has been routinely used to successfully solve a diversity of practical problems in which one or more of the constraints involved is non-convex. As a first step toward addressing this deficiency, we provide convergence results for a prototypical non-convex two-set scenario in which one of the sets is the Euclidean sphere.

We consider the convexity of chance constraints with random right-hand side. While this issue is well understood (thanks to Prekopa’s Theorem) if the mapping operating on the decision vector is componentwise concave, things become more delicate when relaxing the concavity property. In an earlier paper, the significantly weaker r-concavity concept could be exploited, in order to derive eventual convexity (starting from a certain probability level) for feasible sets defined by chance constraints. This result heavily relied on the assumption of the random vector having independent components. A generalization to arbitrary multivariate distributions is all but straightforward. The aim of this chapter is to derive the same convexity result for distributions modeled via copulae. In this way, correlated components are admitted, but a certain correlation structure is imposed through the choice of the copula. We identify a class of copulae admitting eventually convex chance constraints.

Now that we have a solid background in Euclidean geometry, we can go deeper into our study of convex sets begun in Chapter 3. This chapter is devoted to a thorough study of separating and supporting hyperplanes. We prove two geometric versions of the Hahn–Banach theorem, from which we derive separation results for various kinds of pairs of convex sets (open, closed, compact). We prove various versions of Farkas’s lemma, a basic result in the theory of linear programming. We also discuss supporting hyperplanes and prove an important proposition due to Minkowski.

In this chapter, we begin by proving the Brouwer and the Schauder fixed-point theorems. Then we turn to results on homeomorphisms of convex sets and spaces. We prove Keller’s theorem on homeomorphism of infinite-dimensional compact convex sets in Banach spaces to \({\mathbb I}^{{\mathbb N}}\). We also prove the Kadec theorem on the homeomorphism of every separable reflexive space to a Hilbert space. Then we prove some results on uniform, in particular Lipschitz, homeomorphisms.

Let K be a convex body (i.e., a compact convex set with nonempty interior) in d-dimensional Euclidean space \(\mathbb{E}^d\), d ≥ 2. Then the Hadwiger number H(K) of K is the largest number of non-overlapping translates of K that can all touch K. An elegant observation of Hadwiger [154] is the following.

Let K be a convex body (i.e., a compact convex set with nonempty interior) in the d-dimensional Euclidean space \(\mathbb{E}^d\), d ≥ 2. According to Hadwiger [155] an exterior point p ℇ \(\mathbb{E}^d\) \ K of K illuminates the boundary point q of K if the haline emanating from p passing through q intersects the interior of K (at a point not between p and q). Furthermore, a family of exterior points of K say, p1; p2;…; pn illuminates K if each boundary point of K is illuminated by at least one of the point sources p1; p2;…; pn. Finally, the smallest n for which there exist n exterior points of K that illuminate K is called the illumination number of K denoted by I(K). In 1960, Hadwiger [155] raised the following amazingly elementary, but very fundamental question. An equivalent but somewhat different-looking concept of illumination was introduced by Boltyanski in [78]. There he proposed to use directions (i.e., unit vectors) instead of point sources for the illumination of convex bodies. Based on these circumstances we call the following conjecture the Boltyanski-Hadwiger Illumination Conjecture.

Let P be the uniform probability law on the unit cube I d in d dimensions, and P n the corresponding empirical measure. For various classes С of sets A ⊂ I d , upper and lower bounds are found for the probable size of sup {|P n − P)(A)|: A ∈ С}. If С is the collection of lower layers in I 2, or of convex sets in I 3, an asymptotic lower bound is $$\left( {{{\left( {\log n} \right)} \mathord{\left/ {\vphantom {{\left( {\log n} \right)} n}} \right.} n}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right)\left( {\log \log n} \right)^{ - \delta - {1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}} \quad {\rm{for}}\,{\rm{any}}\,\delta > 0.$$ Thus the law of the iterated logarithm fails for these classes. If α > 0, β is the greatest integer 0. When α = d − 1 the same lower bound is obtained as for the lower layers in I 2 or convex sets in I 3. For \(0 < \,\alpha \,\underline \le \,d\, - \,1\) there is also an upper bound equal to a power of log n times the lower bound, so the powers of n are sharp.

An algorithm for finding the intersection of the convex hulls of two sets consisting of finitely many points each is proposed. The problem is modelled by means of a quasidifferentiable (in the sense of Demyanov and Rubinov) optimization problem, which is solved by a descent method for quasidifferentiable functions.

We present necessary and sufficient conditions for identifying global weak minimizers of non-convex multi-objective quadratic optimization problems. We derive these results by exploiting the hidden convexity of the joint range of (non-convex) quadratic functions. We also present numerical examples to illustrate our results.

The Mordukhovich subdifferential, being highly important in variational and nonsmooth analysis and optimization, often happens to be hard to calculate. We propose a method for computing the Mordukhovich subdifferential of differences of sublinear (DS) functions applying the directed subdifferential of differences of convex (DC) functions. We restrict ourselves to the two-dimensional case mainly for simplicity of the proofs and for the visualizations. The equivalence of the Mordukhovich symmetric subdifferential (the union of the corresponding subdifferential and superdifferential) to the Rubinov subdifferential (the visualization of the directed subdifferential) is established for DS functions in two dimensions. The Mordukhovich subdifferential and superdifferential are identified as parts of the Rubinov subdifferential. In addition, the Rubinov subdifferential may be constructed as the Mordukhovich one by Painleve–Kuratowski outer limits of Frechet subdifferentials. The results are applied to the case of DC functions. Examples illustrating the obtained results are presented. 2010 Mathematics Subject Classification. Primary 49J52; Secondary 26B25, 49J50, 90C26

The previous sections indicate a good deal of geometry on unions and intersections of balls that is worthwhile studying. In particular, when we restrict our attention to intersections of balls the underlying convexity suggests a broad spectrum of new analytic and combinatorial results. To make the setup ideal for discrete geometry from now on we look at intersections of finitely many congruent closed d-dimensional balls with non-empty interior in \(\mathbb{E}^d\). In fact, one may assume that the congruent d-dimensional balls in question are of unit radius; that is, they are unit balls of \(\mathbb{E}^d\). Also, it is natural to assume that removing any of the unit balls defining the intersection in question yields the intersection of the remaining unit balls becoming a larger set. Id d = 2, then we call the sets in question disk-polygons and for d ≥ 3 ther are called ball-polyhedra. This definition along with some basic properties of ball-polyhedra (resp., disk-polygons) were introduced by the author in a sequence of talks at the University of Calgary in the fall of 2004. Based on that, the paper [69] written by the author, Langi, Naszodi, and Papez systematically extended those investigations to get a better understanding of the geometry of ballpolyhedra (resp., disk-polygons) by proving a number of theorems, which one can regard as the analogues of the classical theorems on convex polytopes.

In this chapter, we probe several topics that use significant ideas from convexity theory and that have significant applications in various fields. In particular, we prove theorems of Radon, Helly, Kirchberger, Barany, and Tverberg on the combinatorial structure of convex sets, application of Helly’s theorem to semi-infinite programming, in particular to Chebyshev’s approximation problem, homogeneous convex functions, and their applications to inequalities, attainment of optima in maximization of convex functions, decompositions of convex cones, and finally the relationship between the norms of a homogeneous polynomial and its associated symmetric form. The last result has an immediate application to self-concordant functions in interior-point algorithms.

Let f: [0; 1] → ∝ be a function such that f is positive and monotone increasing on (0; 1]; moreover, f(x) = (g(x)) k for some concave function g : [0; 1] → ∝, where k is a positive integer. Then $$F(y) :=\frac{1}{f(y)}\int^y_0 f(x)dx$$ is strictly monotone increasing on (0; 1].

For any given partial ordering \(\preceq\)of a setX, real-valued functionsf1 defined onXwhich satisfyf(x)= f(y) wheneverx_yare var- 2 iously referred to as “monotonic,” “isotonic,” or “order-preserving.” 3 For the ordering of majorization, the order-preserving functions were 4 first systematically studied by I. Schur (1923). In Schur’s honor, such 5 functions are said to be “convex in the sense of Schur,” “Schur-convex,” 6 or “S-convex.” The historical origin of these terms is described in 7 Section 1.C.

In this section we discuss (without proof) some properties of convex figures in the plane. A figure M is called convex if, together with every two points a, b, it contains the whole segment [a, b], (Figure 13.1). Triangles, parallelograms, and trapezoids are convex (Figure 13.2). Disks, sectors with a central angle not exceeding 180°, and segments of disks are also convex (Figure 13.3); so is any segment and any single point. Some nonconvex figures are shown in Figure 13.4.

The problem to find a best solution within the set of optimal solutions of a convex optimization problem is modeled as a bilevel programming problem. It is shown that regularity conditions like Slater’s constraint qualification are never satisfied for this problem. If the lower-level problem is replaced with its (necessary and sufficient) optimality conditions, it is possible to derive a necessary optimality condition for the resulting problem. An example is used to show that this condition in not sufficient even if the initial problem is a convex one. If the lower-level problem is replaced using its optimal value, it is possible to obtain an optimality condition that is both necessary and sufficient in the convex case.

We obtain a series of Hardy type inequalities for domains involving both distance to the boundary and distance to the origin. In particular, we obtain the Hardy─Sobolev inequality for the class of symmetric functions in a ball and prove that for d ≥ 3 the Hardy inequality involving the distance to the boundary holds with the constant 1/4 in a large family of domains not necessarily convex. We also present an example showing that for any positive fixed constant there is an ellipsoid layer such that the Hardy inequality with the distance to the boundary fails.