Caratheodory's Theorem Research Articles

Carathéodory, Helly and the Others in the Max-Plus World

Caratheodory’s, Helly’s and Radon’s theorems are three basic results in discrete geometry. Their max-plus or tropical analogues have been proved by various authors. We show that more advanced results in discrete geometry also have max-plus analogues, namely, the colorful Caratheodory theorem and the Tverberg theorem. A conjecture connected to the Tverberg theorem—Sierksma’s conjecture—although still open for the usual convexity, is shown to be true in the max-plus setting.

Intersecting Families on Sets and the Topology of Cones in Economics

Two classical problems in economics, the existence of a market equilibrium and the existence of social choice functions, are formalized here by the properties of a family of cones associated with the economy. It was recently established that a necessary and sufficient condition for solving the former is the nonempty intersection of the family of cones, and one such condition for solving the latter is the acyclicity of the unions of its subfamilies. We show an unexpected but clear connection between the two problems by establishing a duality property of the homology groups of the nerve defined by the family of cones. ln particular, we prove that the intersection of the family of cones is nonempty if and only if every subfamily has acyclic unions, thus identifying the two conditions that solve the two economic problems. ln addition to their applications to economics, the results arc shown to extend significantly several classical theorems, providing unified and simple proofs: Helly's theorem, Caratheodory's representation theorem, the Knaster-Kuratowski-Marzukiewicz theorem, Brouwer's fixed point theorem, and Leray's theorem on acyclic covers.

Points surrounding the origin

Suppose d > 2, n > d+1, and we have a set P of n points in d-dimensional Euclidean space. Then P contains a subset Q of d points such that for any p ∈ P, the convex hull of Q∪{p} does not contain the origin in its interior. We also show that for non-empty, finite point sets A 1, ..., A d+1 in ℝ d , if the origin is contained in the convex hull of A i ∪ A j for all 1≤i<j≤d+1, then there is a simplex S containing the origin such that |S∩A i |=1 for every 1≤i≤d+1. This is a generalization of Barany’s colored Caratheodory theorem, and in a dual version, it gives a spherical version of Lovasz’ colored Helly theorem.

Borel–Carathéodory Type Theorem for Monogenic Functions

In this paper we give a generalization of the classical Borel–Caratheodory theorem in complex analysis to higher dimensions in the framework of Quaternionic Analysis.

Convexity in Topological Affine Planes

We extend to topological affine planes the standard theorems of convexity, among them the separation theorem, the anti-exchange theorem, Radon's, Helly's, Caratheodory's, and Kirchberger's theorems, and the Minkowski theorem on extreme points. In a few cases the proofs are obtained by adapting proofs of the original results in the Euclidean plane; in others it is necessary to devise new proofs that are valid in the more general setting considered here.

Optimal discrimination designs for exponential regression models

We investigate optimal designs for discriminating between exponential regression models of different complexity, which are widely used in the biological sciences; see, e.g., Landaw [1995. Robust sampling designs for compartmental models under large prior eigenvalue uncertainties. Math. Comput. Biomed. Appl. 181–187] or Gibaldi and Perrier [1982. Pharmacokinetics. Marcel Dekker, New York]. We discuss different approaches for the construction of appropriate optimality criteria, and find sharper upper bounds on the number of support points of locally optimal discrimination designs than those given by Caratheodory's Theorem. These results greatly facilitate the numerical construction of optimal designs. Various examples of optimal designs are then presented and compared to different other designs. Moreover, to protect the experiment against misspecifications of the nonlinear model parameters, we adapt the design criteria such that the resulting designs are robust with respect to such misspecifications and, again, provide several examples, which demonstrate the advantages of our approach.

Colourful Simplicial Depth

Inspired by Barany's Colourful Caratheodory Theorem, we introduce a colourful generalization of Liu's simplicial depth. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any d-dimensional configuration is d2 + 1 and that the maximum is dd+1 + 1. We exhibit configurations attaining each of these depths, and apply our results to the problem of bounding monochrome (non-colourful) simplicial depth.

The Hadamard Theorem and Borel–Carathéodory Theorem on Riemann Surfaces

In this paper, we first define a kind of pseudo–distance function and annulus domain on Riemann surfaces, then prove the Hadamard Theorem and the Borel–Caratheodory Theorem on any Riemann surfaces.

Noncommutative sums of squares

We present a proof of Helton?s sum-of-squares theorem based upon a theorem of Caratheodory and a Hahn?Banach separation argument.

Approximation of a Metric on a Dominating Cone

A theorem on the representation of a pseudoball of a dominating cone is proven. This theorem is an analog of the Caratheodory theorem on the representation of convex sets. The theorem allows to develop numerical methods for quantitative evaluation of systems in many indicators.

Existence and Relaxation for Finite-Dimensional Optimal Control Problems Driven by Maximal Monotone Operators

We study the optimal control of a class of nonlinear finite-dimensional optimal control problems driven by a maximal monotone operator which is not necessarily everywhere defined. So our model problem incorporates systems monitored by variational inequalities. First we prove an existence theorem using the reduction method of Berkovitz and Cesari. This requires a convexity hypothesis. When this convexity condition is not satisfied, we have to pass to an augmented, convexified problem known as the "relaxed problem". We present four relaxation methods. The first uses Young measures, the second uses multi-valued dynamics, the third is based on Caratheodory's theorem for convex sets in RN and the fourth uses lower semicontinuity arguments and Γ-limits. We show that they are equivalent and admissible, which roughly speaking means that the corresponding relaxed problem is in a sense the "closure" of the original one.

Cell complexes, oriented matroids and digital geometry

Abstract cell complexes (ACCs) were introduced by Kovalevsky as a means of solving certain connectivity paradoxes in graph-theoretic digital topology, and to this extent provide an improved theoretical basis for image analysis. In this work we argue that ACCs are a very natural setting for digital geometry, to the extent that their use permits simple, almost trivial formulations of major convexity results, including Caratheodory's, Helly's and Radon's theorems. We also discuss the relevance of oriented matroids to digital geometry.

Integral decomposition of polyhedra and some applications in mixed integer programming

This paper addresses the question of decomposing an infinite family of rational polyhedra in an integer fashion. It is shown that there is a finite subset of this family that generates the entire family. Moreover, an integer analogue of Caratheodory's theorem carries over to this general setting. The integer decomposition of a family of polyhedra has some applications in integer and mixed integer programming, including a test set approach to mixed integer programming.

Asymptotic behavior of semigroups of ρ-non-expansive and holomorphic mappings on the Hilbert Ball

We study generated semigroups of those self-mappings of the Hilbert ball which are non-expansive with respect to the hyperbolic metric. We find optimal convergence rates for such semigroups to interior stationary and boundary sink points. Since the hyperbolic metric is not defined on the boundary, the usual approach treats these two cases separately. In contrast with this practice, we use a special non-Euclidean “distance” (which induces the original topology) to present a unified theory. Our approach leads to new results even in the one-dimensional case. When the semigroups consist of holomorphic self-mappings, we obtain the rather unexpected phenomenon of universal rates of convergence of an exponential type. In particular, in the case of a boundary sink point we establish a continuous analog of the celebrated Julia–Wolff–Caratheodory theorem.

A proof of the basic measure extension theorem of Carathéodory

In this note a direct elementary proof of Caratheodory's measure extension theorem is presented. It is based on an approximation argument for outer measures where elements of the \(\sigma\)-algebra are approached by elements of the underlying algebra of sets with respect to the symmetric difference.

The dual parameterization approach to optimal least square FIR filter design subject to maximum error constraints

This paper is concerned with the design of linear-phase finite impulse response (FIR) digital filters for which the weighted least square error is minimized, subject to maximum error constraints. The design problem is formulated as a semi-infinite quadratic optimization problem. Using a newly developed dual parameterization method in conjunction with the Caratheodory's dimensional theorem, an equivalent dual finite dimensional optimization problem is obtained. The connection between the primal and the dual problems is established. A computational procedure is devised for solving the dual finite dimensional optimization problem. The optimal solution to the primal problem can then be readily obtained from the dual optimal solution. For illustration, examples are solved using the proposed computational procedure.

Simple proofs of some fundamental properties of the Julia set

Let $f$ be a holomorphic self-map of $\mathbb{C} \backslash \{ 0 \}, \mathbb{C}$, or the extended complex plane $\overline{\mathbb{C}}$ that is neither injective nor constant. This paper gives new and elementary proofs of the well-known fact that the Julia set of $f$ is a non-empty perfect set and coincides with the closure of the set of repelling cycles of $f$. The proofs use Montel–Caratheodory's theorem but do not use results from Nevanlinna theory.

A dual approach to continuous-time envelope-constrained filter design via orthonormal filters

The envelope-constrained (EC) filtering problem was initially posed in the continuous-time domain as a constrained L/sup 2/ space optimization problem. However, only the discretized version has been solved, using various approaches. In this paper we consider the continuous-time EC filtering problem formulated using an orthonormal basis defined on a Hilbert vector space. Caratheodory's theorem and the dual parametrization method are used to obtain a dual finite-dimensional optimization problem. Two iterative algorithms based on the gradient flow technique are then developed for solving this dual problem. The design of an EC optimal equalization filter for a communication channel is solved to illustrate the effectiveness of the proposed algorithm.

Aircraft Trajectory Optimization in the Horizontal Plane

Abstract The problem of minimum time trajectory optimization for aircraft in the presence of winds is considered. The solutions are obtained by employing a parametrization technique based on the famous theorem of Caratheodory in order to reduce the problem into a finite dimensional optimization problem, without using the minimum principle or similar analysis methods. Numerical solutions are presented for cases with and without the presence of winds in order to demonstrate the effect of winds on the trajectory planning.

Aircraft Trajectory Optimization in the Horizontal Plane

The problem of minimum time trajectory optimization for aircraft in the presence of winds is considered. The solutions are obtained by employing a parametrization technique based on the famous theorem of Caratheodory in order to reduce the problem into a finite dimensional optimization problem, without using the minimum principle or similar analysis methods. Numerical solutions are presented for cases with and without the presence of winds in order to demonstrate the effect of winds on the trajectory planning.