An Extension of the Ham Sandwich Theorem

Živaljević, Rade T.; Vrećica, Siniša T.

doi:10.1112/blms/22.2.183

Cited by 51 publications

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“…Establishing the relation between Rado's theorem on general measure (see [11]) and the Ham sandwich theorem, the following result is proved in [21] stating that these two results belong to the same family. Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

Tverberg's Conjecture

Vrećica¹

2003

Discrete and Computational Geometry

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In 1989 Helge Tverberg proposed a quite general conjecture in Discrete Geometry, which could be considered as the common basis for many results in Combinatorial Geometry, and at the same time as a discrete analogue of the common transversal theorems. It implies or contains as special cases many classical "coincidence" results such as Radon's theorem, Rado's theorem, the Ham sandwich theorem, "non-embeddability" results (e.g. non-embeddability of graphs K 5 and K 3,3 in R 2 ), etc. The main goal of this short note is to verify this conjecture in one new, non-trivial case. We obtain the continuous version of the conjecture. So, it is not surprising that we use topological methods, or more precisely the methods of equivariant topology and the theory of characteristic classes.

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Section: Introductionmentioning

confidence: 99%

Tverberg's Conjecture

Vrećica¹

2003

Discrete and Computational Geometry

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“…A cohomological proof of this observation can be found in [5], see also [20], while an alternative proof, based on Schubert calculus, is in [8], section 1.5. From here we deduce that θ k+1 = w k n−k θ = 0 which follows from w k n−k = 0.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…n } with the topology inherited from the product space Gr n,n−k × R n , and its projection is (L [5] or [20], Proposition 2, any k continuous cross-sections of this bundle have a common zero. In other words, there exists a plane…”

Section: Proof Of Theoremmentioning

confidence: 99%

Plane Sections of Convex Bodies of Maximal Volume

Makai¹,

Vrećica²,

Živaljević³

2001

Discrete Comput Geom

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Abstract. Let K = {K 0 , ..., K k } be a family of convex bodies in R n , 1 ≤ k ≤ n−1. We prove, generalizing results from [9], [10], [13], [14], that there always exists an affine k-dimensional plane A k ⊆ R n , called a common maximal k-transversal of K, such that for each i ∈ {0, ..., k} and eachof convex bodies in R n , l < k, the set C k (K) of all common maximal ktransversals of K is not only non-empty but has to be "large" both from the measure theoretic and the topological point of view. It is shown that C k (K) cannot be included in a ν-dimensional C 1 submanifold (or more generally in an (H ν , ν)-rectifiable, H ν -measurable subset) of the affine Grassmannian AGr n,k of all affine k-dimensional planes of R n , of O(n + 1)-invariant ν-dimensional (Hausdorff) measure less than some positive constant c n,k,l , where ν = (k − l)(n − k). As usual, the "affine" Grassmannian AGr n,k is viewed as a subspace of the Grassmannian Gr n+1,k+1 of all linear (k + 1)-dimensional subspaces of R n+1 . On the topological side we show that there exists a nonzero cohomology class θ ∈ H n−k (G n+1,k+1 ; Z 2 ) such that the class θ l+1 is concentrated in an arbitrarily small neighborhood of C k (K). As an immediate consequence we deduce that the Lyusternik-Shnirel'man category of the space C k (K) relative to Gr n+1,k+1 is ≥ k − l. Finally, we show that there exists a link between these two results by showing that a cohomologically "big" subspace of Gr n+1,k+1 has to be large also in a measure theoretic sense.

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“…Živaljević and Vrećica [26] give a unification of Rado's Theorem with the Ham Sandwich Theorem: For measures μ 1 , . .…”

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confidence: 98%

Uneven Splitting of Ham Sandwiches

Breuer

2009

Discrete Comput Geom

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Let m_1,...,m_n be continuous probability measures on R^n and a_1,...,a_n in [0,1]. When does there exist an oriented hyperplane H such that the positive half-space H^+ has m_i(H^+)=a_i for all i in [n]? It is well known that such a hyperplane does not exist in general. The famous ham sandwich theorem states that if a_i=1/2 for all i, then such a hyperplane always exists. In this paper we give sufficient criteria for the existence of H for general a_i in [0,1]. Let f_1,...,f_n:S^{n-1}->R^n denote auxiliary functions with the property that for all i the unique hyperplane H_i with normal v that contains the point f_i(v) has m_i(H_i^+)=a_i. Our main result is that if Im(f_1),...,Im(f_n) are bounded and can be separated by hyperplanes, then there exists a hyperplane H with m_i(H^+)=a_i for all i. This gives rise to several corollaries, for instance if the supports of m_1,...,m_n are bounded and can be separated by hyperplanes, then H exists for any choice of a_1,...,a_n in [0,1]. We also obtain results that can be applied if the supports of m_1,...,m_n overlap.Comment: 18 pages, 2 figure

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An Extension of the Ham Sandwich Theorem

Cited by 51 publications

References 0 publications

Tverberg's Conjecture

Tverberg's Conjecture

Plane Sections of Convex Bodies of Maximal Volume

Uneven Splitting of Ham Sandwiches

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