Radon's Theorem Research Articles

A Radon theorem for Helly graphs

Etude et caracterisation des graphes de Helly (retracts absolus de graphes reflexifs). Etablissement du theoreme de Radon. Determination du nombre de Radon

Separation theorems for oriented matroids

In this paper we show that Minty's lemma can be used to prove the Hahn-Banach theorem as well as other theorems in this class such as Radon's and Helly's theorem for oriented matroids having an intersection property which guarantees that every pair of flats intersects in some point extension O ∪ p of the oriented matroid O .

Maximum entropy Fourier synthesis with application to diffraction tomography

In diffraction tomography, the generalized Radon theorem relates the Fourier transform (FT) of the diffracted field to the two-dimensional FT of the diffracting object. The relationship stands on algebraic contours, which are semicircles in the case of Born or Rytov first-order linear approximations. But the corresponding data are not sufficient to determine uniquely the solution. We propose a maximum entropy method to reconstruct the object from either the Fourier domain data or directly from the original diffracted field measurements. To do this, we give a new definition for the entropy of an object considered as a function of R(2) to C. To take into account the presence of noise, a chi-squared statistic is added to the entropy measure. The objective function thus obtained is minimized using variational techniques and a conjugate-gradient iterative method. The computational cost and practical implementation of the algorithm are discussed. Some simulated results are given which compare this new method with the classical ones.

Helly-type theorems for pseudoline arrangements in P 2

We prove duals of Radon's theorem, Helly's theorem, Carathéodory's theorem, and Kirchberger's theorem for arrangements of pseudolines in the real projective plane, which generalize the original versions of those theorems for plane configurations of points. We also prove a topological generalization of the pseudoline-dual of Helly's theorem.

A generalization of Radon's theorem II

A new proof is given of the following result: Let m and d be positive integers, and let a set of md + m − d points be given in d-dimensional space. Then the set can be partitioned into m sets such that the m convex polytopes spanned by the sets have a non-empty intersection.

On a common generalization of Borsuk's and Radon's theorem

On a common generalization of Borsuk's and Radon's theorem

Intersections of spherically convex sets

New theorems of Helly type are proved concerning the intersection of convex cones with a common vertex or, equivalently, the intersection of sets on a sphere which are convex in the sense of Robinson. The proofs of these theorems are based on a lemma which is a spherical analog and generalization of Radon's theorem.

The Geometry of Radon's Theorem

(1972). The Geometry of Radon's Theorem. The American Mathematical Monthly: Vol. 79, No. 9, pp. 949-963.

The Geometry of Radon's Theorem

The Geometry of Radon's Theorem

A theorem on families of sets

We prove the following result and transfinite extensions of it: Let ( M i : i ϵ I) be a family of non-zero subsets of the set S. If the cardinalities | I| = f and | S| = n are finite and f > n( r − 1), then one can find r disjoint subsets I υ ( υ = 1,…, r) of I for which ⋃ i∈I 1 M i = … = ⋃ i∈I r M i The proof is constructive. We apply a generalization by R. Rado of P. Hall's celebrated theorem on systems of representatives. Another proof of the above result has been found by H. Tverberg (see [3]). Tverberg applies his generalization of Radon's theorem (see [2]). He also shows by an example that the result is in a sense best possible.

Neighbourliness and Radon's theorem

Let X be a finite set of points in E d . Then a partition of X into two non-empty subsets X 1 and X 2 ( X 1 ∪ X 2 = X , X 1 ∩ X 2 = ∅) will be called a Radon partition if .

An extension of Radon's theorem

An extension of Radon's theorem

A Further Generalization of Radon's Theorem

A Further Generalization of Radon's Theorem

A Generalization of Radon's Theorem

Journal of the London Mathematical SocietyVolume s1-41, Issue 1 p. 123-128 Notes and papers A Generalization of Radon's Theorem H. Tverberg, H. Tverberg University of Bergen Bergen, NorwaySearch for more papers by this author H. Tverberg, H. Tverberg University of Bergen Bergen, NorwaySearch for more papers by this author First published: 1966 https://doi.org/10.1112/jlms/s1-41.1.123Citations: 165AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volumes1-41, Issue11966Pages 123-128 RelatedInformation

The representation of linear operators from 𝐿^{𝑝} to 𝐿

A satisfactory representation theory in terms of integrals for linear operators from an LP space to an arbitrary Banach space X has not as yet been developed for p>1. For the case p=1, Dunford and Pettis [3] have obtained results provided X satisfies certain rather general restrictions. In particular, the representations for the cases in which X is an L space or an LP space are well known. The case in which the domain is L and X is an arbitrary space has been treated by Rickart [6] who obtained a representation in terms of the Phillips-Birkhoff integral [5; 2 ]. If the range space X is a function space in which the values of the functions at a fixed point in their common domain of definition yield a linear functional over the space, the problem can frequently be treated if the form of the linear functional over the domain space is known by considering the operator as a family of linear functionals over the space. In the present paper we use essentially this technique to obtain a representation for operators from LP to L in terms of the Lebesgue integral. We denote by (R, cS), (S, y) measure spaces where R, S are sets, Xk, Py completely additive measures defined over o-rings 7(R), 7(S) of subsets in R, S respectively, and we assume that R, S are unions of countable families of sets of finite measure. The set of all realvalued functions {x(t) } defined over R which are measurable and such that I x(t) P' is integrable over R with respect to qk is denoted by LP(R, qk). It is well known that if 1 1, p' defined by 1/p+1/p' = 1 and h ELI(R, c5) if p = 1. We shall make extensive use of the theorem of Radon and Nikodym [8] which states that L(R, qk) is isometrically isomorphic to the space of completely additive and absolutely continuous set functions A(R, c5) defined over 7(R) where FCA(R, c5) implies that ||F||