Ball and spindle convexity with respect to a convex body

Let ${C \subset \mathbb{R}^n}$ be a convex body. We introduce two notions of convexity associated to C. A set K is C-ball convex if it is the intersection of translates of C, or it is either ${\emptyset}$ , or ${\mathbb{R}^n}$ . The C-ball convex hull of two points is called a C-spindle. K is C-spindle convex if it contains the C-spindle of any pair of its points. We investigate how some fundamental properties of conventional convex sets can be adapted to C-spindle convex and C-ball convex sets. We study separation properties and Carathéodory numbers of these two convexity structures. We investigate the basic properties of arc-distance, a quantity defined by a centrally symmetric planar disc C, which is the length of an arc of a translate of C, measured in the C-norm that connects two points. Then we characterize those n-dimensional convex bodies C for which every C-ball convex set is the C-ball convex hull of finitely many points. Finally, we obtain a stability result concerning covering numbers of some C-ball convex sets, and diametrically maximal sets in n-dimensional Minkowski spaces.     

Equipartitioning by a convex 3-fan

We show that for a given planar convex set K of positive area there exist three pairwise internally disjoint convex sets whose union is K such that they have equal area and equal perimeter.     

Infeasibility analysis for systems of quadratic convex inequalities

In this paper we address the problem of the infeasibility of systems defined by quadratic convex inequality constraints. In particular, we investigate properties of irreducible infeasible sets and provide an algorithm that identifies a set of all constraints (K) that may affect the feasibility status of the system after some perturbation of the right-hand sides. We show that all irreducible sets, as well as infeasibility sets, are subsets of the set K, and that every infeasible system contains an inconsistent subsystem of cardinality not greater than the number of variables plus one. The results presented in this paper are also applicable to linear systems.     

Reduced convex bodies in Euclidean space—A survey

A convex body R in Euclidean space E^d is called reduced if the minimal width Δ(K) of each convex body K⊂R different from R is smaller than Δ(R). This definition yields a class of convex bodies which contains the class of complete sets, i.e., the family of bodies of constant width. Other obvious examples in E² are regular odd-gons. We know a relatively large amount on reduced convex bodies in E². Besides theorems which permit us to understand the shape of their boundaries, we have estimates of the diameter, perimeter and area. For d≥3 we do not even have tools which permit us to recognize what the boundary of R looks like. The class of reduced convex bodies has interesting applications. We present the current state of knowledge about reduced convex bodies in E^d, recall some striking related research problems, and put a few new questions.     

Small set in a large box

Let K ? ?^d be a compact convex set which is an intersection of half-spaces defined by at most two coordinates. Let Q be the smallest axes-parallel box containing K. We show that, as the dimension d grows, the ratio diam Q/ diam K can be arbitrarily large. We also give examples of compact sets in Banach spaces which are not contained in any compact contractive set.     

Infinite dimensional convex sets with a property common to finite dimensional convex sets

We characterize those metrizable compact convex sets K that contain a Bauer simplex B such that any two affine continuous functions on K coinciding on B are equal; further, we show that all metrizable simplexes and dual balls of separable L¹-preduals have this property.     

Convex polarities over ordered fields

We use tools and methods from real algebraic geometry (spaces of ultrafilters, elimination of quantifiers) to formulate a theory of convexity in K^N over an arbitrary ordered field. By defining certain ideal points (which can be viewed as generalizations of recession cones) we obtain a generalized notion of polar set. These satisfy a form of polar duality that applies to general convex sets and does not reduce to classical duality if K is the field of real numbers. As an application we give a partial classification of total orderings of Artinian local rings and two applications to ordinary convex geometry over the real numbers.     

Kirchberger-type theorems for separation by convex domains

We say that a convex set K in ? ^d strictly separates the set A from the set B if A ? int(K) and B ? cl K = ø. The well-known Theorem of Kirchberger states the following. If A and B are finite sets in ? ^d with the property that for every T ? A?B of cardinality at most d + 2, there is a half space strictly separating T ? A and T ? B, then there is a half space strictly separating A and B. In short, we say that the strict separation number of the family of half spaces in ? ^d is d + 2.In this note we investigate the problem of strict separation of two finite sets by the family of positive homothetic (resp., similar) copies of a closed, convex set. We prove Kirchberger-type theorems for the family of positive homothets of planar convex sets and for the family of homothets of certain polyhedral sets. Moreover, we provide examples that show that, for certain convex sets, the family of positive homothets (resp., the family of similar copies) has a large strict separation number, in some cases, infinity. Finally, we examine how our results translate to the setting of non-strict separation.     

On convex bodies containing a given number of lattice points

Given a convex body K in Euclidean space, a necessary and sufficient condition is established in order that for each n there exists a homothetic copy of K containing exactly n lattice points. Similar theorems are proved for congruent copies of K and for discrete sets other than lattices.     

Iterated Boolean random varieties and application to fracture statistics models

Models of random sets and of point processes are introduced to simulate some specific clustering of points, namely on random lines in R² and R³ and on random planes in R³. The corresponding point processes are special cases of Cox processes. The generating distribution function of the probability distribution of the number of points in a convex set K and the Choquet capacity T (K) are given. A possible application is to model point defects in materials with some degree of alignment. Theoretical results on the probability of fracture of convex specimens in the framework of the weakest link assumption are derived, and used to compare geometrical effects on the sensitivity of materials to fracture.     

Motzkin predecomposable sets

We introduce and study the family of sets in a finite dimensional Euclidean space which can be written as the Minkowski sum of a compact and convex set and a convex cone (not necessarily closed). We establish several properties of the class of such sets, called Motzkin predecomposable, some of which hold also for the class of Motzkin decomposable sets (i.e., those for which the convex cone in the decomposition is requested to be closed), while others are specific of the new family.     

A convex sets in general position

The notion of convex cones in general position has turned out to be useful in convex programming theory. In this paper we extend the notion to convex sets and give some characterizations which yield a better insight into this concept. We also consider the case of convex sets in S-general position.     

On maximal convex lattice polygons inscribed in a plane convex set

Given a convex compact setK ? ?₂ what is the largestn such thatK contains a convex latticen-gon? We answer this question asymptotically. It turns out that the maximaln is related to the largest affine perimeter that a convex set contained inK can have. This, in turn, gives a new characterization ofK ₀, the convex set inK having maximal affine perimeter.     

On stable set polyhedra for K1,3-free graphs

We give several classes of facets for the convex hull of incidence vectors of stable sets in a K_1,3-free graph, including facets with (a, a + 1)-valued coefficients, where a = 1, 2, 3,…. These provide counterexamples to three recent conjectures concerning such facets. We also give a necessary and sufficient condition for a minimal imperfect graph to be an odd hole or an odd antihole and indicate that minimal imperfect K_1,3-free graphs satisfy the condition.     

Sequential optimality conditions for composed convex optimization problems

Using a general approach which provides sequential optimality conditions for a general convex optimization problem, we derive necessary and sufficient optimality conditions for composed convex optimization problems. Further, we give sequential characterizations for a subgradient of the precomposition of a K-increasing lower semicontinuous convex function with a K-convex and K-epi-closed (continuous) function, where K is a nonempty convex cone. We prove that several results from the literature dealing with sequential characterizations of subgradients are obtained as particular cases of our results. We also improve the above mentioned statements.     

Decompositions,Partitions, and Coverings with Convex Polygons and Pseudo-Triangles

We propose a novel subdivision of the plane that consists of both convex polygons and pseudo-triangles. This pseudo-convex decomposition is significantly sparser than either convex decompositions or pseudo-triangulations for planar point sets and simple polygons. We also introduce pseudo-convex partitions and coverings. We establish some basic properties and give combinatorial bounds on their complexity. Our upper bounds depend on new Ramsey-type results concerning disjoint empty convex k-gons in point sets.     

Algebraic signatures of convex and non-convex codes

A convex code is a binary code generated by the pattern of intersections of a collection of open convex sets in some Euclidean space. Convex codes are relevant to neuroscience as they arise from the activity of neurons that have convex receptive fields. In this paper, we develop algebraic methods to determine if a code is convex. Specifically, we use the neural ideal of a code, which is a generalization of the Stanley–Reisner ideal. Using the neural ideal together with its standard generating set, the canonical form, we provide algebraic signatures of certain families of codes that are non-convex. We connect these signatures to the precise conditions on the arrangement of sets that prevent the codes from being convex. Finally, we also provide algebraic signatures for some families of codes that are convex, including the class of intersection-complete codes. These results allow us to detect convexity and non-convexity in a variety of situations, and point to some interesting open questions.     

Decomposing Rd into finitely many semigroups

We determine the precise structure of those (additive) semigroups in R^d which belong to at least one partition of R^d into finitely many disjoint Borel measurable semigroups.We also find the structure of the convex sets in R^d which belong to some partition of R^d into finitely many disjoint convex sets.     

A complementarity partition theorem for multifold conic systems

Consider a homogeneous multifold convex conic system $$Ax = 0, \quad x\in K_1\times \cdots \times K_r$$ and its alternative system $$A^t y \in K_1^*\times \cdots \times K_r^*$$ , where K ₁,..., K _r are regular closed convex cones. We show that there is a canonical partition of the index set {1,...,r} determined by certain complementarity sets associated to the most interior solutions to the two systems. Our results are inspired by and extend the Goldman–Tucker Theorem for linear programming.     

Necessary and sufficient conditions for the validity of Jensen’s inequality

We consider the d-dimensional Jensen inequality $$ T[\varphi(f_1, \dots, f_d)]\, \ge \, \varphi(T[f_1], \dots, T[f_d])\quad\quad(\ast)$$ T [ φ ( f 1 , … , f d ) ] ≥ φ ( T [ f 1 ] , … , T [ f d ] ) ( * ) as it was established by McShane in 1937r. Here T is a functional, φ is a convex function defined on a closed convex set ${K\subset \mathbb{R}^d}$ K ? R d , and f ₁, . . . , f _d are from some linear space of functions. Our aim is to find necessary and sufficient conditions for the validity of (*). In particular, we show that if we exclude three types of convex sets K, then Jensen’s inequality holds for a sublinear functional T if and only if T is linear, positive, and satisfies T[1] = 1. Furthermore, for each of the excluded types of convex sets, we present nonlinear, sublinear functionals T for which Jensen’s inequality holds. Thus the conditions on K are optimal. Our contributions generalize or complete several known results.