Papers by Rade Zivaljevic
arXiv (Cornell University), Dec 23, 2004
A well known problem of B. Grünbaum [Grü60] asks whether for every continuous mass distribution (... more A well known problem of B. Grünbaum [Grü60] asks whether for every continuous mass distribution (measure) dµ = f dm on R n there exist n hyperplanes dividing R n into 2 n parts of equal measure. It is known that the answer is positive in dimension n = 3 [Had66] and negative for n ≥ 5, [Avis84] [Ram96]. We give a partial solution to Grünbaum's problem in the critical dimension n = 4 by proving that each measure µ in R 4 admits an equipartition by 4 hyperplanes, provided that it is symmetric with respect to a 2-dimensional, affine subspace L of R 4 . Moreover we show, by computing the complete obstruction in the relevant group of normal bordisms, that without the symmetry condition, a naturally associated topological problem has a negative solution. The computation is based on the Koschorke's exact singularity sequence [Kosch81] and the remarkable properties of the essentially unique, balanced, binary Gray code in dimension 4, [Toot56] [Knu01].
arXiv (Cornell University), Dec 24, 2021
Following a novel approach, where the emphasis is on configuration spaces and equivariant topolog... more Following a novel approach, where the emphasis is on configuration spaces and equivariant topology, we prove several results addressing the envy-free division problem in the presence of an unpredictable (secretive, non-cooperative) player, called the dragon. There are two basic scenarios. 1. There are r -1 players and a dragon. Once the "cake" is divided into r parts, the dragon makes his choice and grabs one of the pieces. After that the players should be able to share the remaining pieces in an envy-free fashion. 2. There are r + 1 players who divide the cake into r pieces. A ferocious dragon comes and swallows one of the players. The players need to cut the cake in advance in such a way that no matter who is the unlucky player swallowed by the dragon, the remaining players can share the tiles in an envy-free manner. We emphasize that in both settings the players are allowed to choose degenerate pieces of the cake. Moreover, the players construct in advance both a cut of the cake and a decision tree, allowing them to minimize the uncertainty of what pieces can be given to each of the players.
Topological Methods in Nonlinear Analysis, Mar 1, 2015
In some recent papers the classical 'splitting necklace theorem' is linked in an interesting way ... more In some recent papers the classical 'splitting necklace theorem' is linked in an interesting way with a geometric 'pattern avoidance problem', see Alon et al. (Proc. Amer. Math. Soc., 2009), Grytczuk and Lubawski (arXiv:1209.1809 [math.CO]), and Lasoń (arXiv:1304.5390v1 [math.CO]). Following these authors we explore the topological constraints on the existence of a (relaxed) measurable coloring of R d such that any two distinct, non-degenerate cubes (parallelepipeds) are measure discernible. For example, motivated by a conjecture of Lasoń, we show that for every collection µ 1 , ..., µ 2d−1 of 2d − 1 continuous finite measures on R d , there exist two nontrivial axis-aligned d-dimensional cuboids (rectangular parallelepipeds) C 1 and C 2 such that µ i (C 1) = µ i (C 2) for each i ∈ {1, ..., 2d − 1}. We also show by examples that the bound 2d − 1 cannot be improved in general. These results are steps in the direction of studying general topological obstructions for the existence of non-repetitive colorings of measurable spaces.
arXiv (Cornell University), Mar 20, 2017
We show that the cyclohedron (Bott-Taubes polytope) W n arises as the polar dual of a Kantorovich... more We show that the cyclohedron (Bott-Taubes polytope) W n arises as the polar dual of a Kantorovich-Rubinstein polytope KR(ρ), where ρ is an explicitly described quasimetric (asymmetric distance function) satisfying strict triangle inequality. From a broader perspective, this phenomenon illustrates the relationship between a nestohedron ∆ F (associated to a building set F) and its non-simple deformation ∆ F , where F is an irredundant or tight basis of F (Definition 21). Among the consequences are a new proof of a recent result of Gordon and Petrov (Arnold Math. J. 3 (2), 205-218 (2017)) about f-vectors of generic Kantorovich-Rubinstein polytopes and an extension of a theorem of Gelfand, Graev, and Postnikov, about triangulations of the type A, positive root polytopes.
Discrete Mathematics, Oct 1, 2013
We give an elementary proof of a formula expressing the rotation number of a cyclic unimodular se... more We give an elementary proof of a formula expressing the rotation number of a cyclic unimodular sequence L = u 1 u 2. .. u d of lattice vectors u i ∈ Z 2 in terms of arithmetically defined local quantities. The formula has been originally derived by A. Higashitani and M. Masuda (arXiv:1204.0088v2 [math.CO]) with the aid of the Riemann-Roch formula applied in the context of toric topology. These authors also demonstrated that a generalized version of the 'Twelve-point theorem' and a generalized Pick's formula are among the consequences or relatives of their result. Our approach emphasizes the role of 'discrete curvature invariants' µ(a, b, c), where {a, b} and {b, c} are bases of Z 2 , as fundamental discrete invariants of modular lattice geometry.
arXiv (Cornell University), Nov 3, 2022
We prove a relative of both the original and the optimal (Type B) version of the Colored Tverberg... more We prove a relative of both the original and the optimal (Type B) version of the Colored Tverberg theorem ofŽivaljević and Vrećica (Theorems 2.6 and 2.7), which modifies these results in two different ways. (1) We extend the original theorems beyond the prime powers by showing that the theorem is valid if the number of rainbow faces is q = p n − 1. (2) The size of some rainbow simplices may be smaller than in the original theorems. More precisely |C i | ∈ {2q − 2, 2q + 1} while (for comparison) in the original theorems it is |C i | = 2q − 1. The proof relies on equivariant index theory and a result of Volovikov [10] about partial coincidences of maps f : X → R d , from a G-space into the Euclidean space.
The coloured Tverberg theorem, extensions and new results
Izvestiya: Mathematics, Apr 1, 2022
We prove amultiple coloured Tverberg theoremand abalanced coloured Tverberg theorem, applying dif... more We prove amultiple coloured Tverberg theoremand abalanced coloured Tverberg theorem, applying different methods, tools and ideas. The proof of the first theorem uses a multiple chessboard complex (as configuration space) and the Eilenberg–Krasnoselskii theory of degrees of equivariant maps for non-free group actions. The proof of the second result relies on the high connectivity of the configuration space, established by using discrete Morse theory.
A reversed Kakutani's fixed point theorem
Mathematical Omnibus: Thirty Lectures on Classic Mathematics
American Mathematical Monthly, 2013
... The reader will find here theorems of Isaac Newton and Leonhard Euler, Augustin Louis Cauchy ... more ... The reader will find here theorems of Isaac Newton and Leonhard Euler, Augustin Louis Cauchy and Carl Gustav Jacob Jacobi, Michel Chasles and Pafnuty Chebyshev, Max Dehn and James Alexander, and many other great mathematicians of the past. ...
A Loeb Measure Approach To The Riesz Representation Theorem
Publications De L'institut Mathematique, 1982
Transactions of the American Mathematical Society, 2008
A well-known problem of B. Grünbaum (1960) asks whether for every continuous mass distribution (m... more A well-known problem of B. Grünbaum (1960) asks whether for every continuous mass distribution (measure) dµ = f dm on R n there exist n hyperplanes dividing R n into 2 n parts of equal measure. It is known that the answer is positive in dimension n = 3 (see H. Hadwiger (1966)) and negative for n ≥ 5 (see D. Avis (1984) and E. Ramos (1996)). We give a partial solution to Grünbaum's problem in the critical dimension n = 4 by proving that each measure µ in R 4 admits an equipartition by 4 hyperplanes, provided that it is symmetric with respect to a 2-dimensional affine subspace L of R 4. Moreover we show, by computing the complete obstruction in the relevant group of normal bordisms, that without the symmetry condition, a naturally associated topological problem has a negative solution. The computation is based on Koschorke's exact singularity sequence (1981) and the remarkable properties of the essentially unique, balanced binary Gray code in dimension 4; see G. C. Tootill (1956) and D. E. Knuth (2001).
Combinatorics of Topological Posets
Advances in Applied Mathematics, Nov 1, 1998
ABSTRACT
The Tverberg-Vrećica problem and the combinatorial geometry on vector bundles
Israel Journal of Mathematics, Dec 1, 1999
It is shown that many classical and many new combinatorial geometric results about finite sets of... more It is shown that many classical and many new combinatorial geometric results about finite sets of points inR d , specially the theorems of Tverberg type, can be generalized to the case of vector bundles, where they become combinatorial geometric statements about finite families of continuous cross-sections. The well known Tverberg-Vrećica conjecture is interpreted as a result of this type
Mitteilungen der Deutschen mathematiker-Vereinigung, 1995
Topological Methods in Nonlinear Analysis, Feb 26, 2023
The classical approach to envy-free division and equilibrium problems arising in mathematical eco... more The classical approach to envy-free division and equilibrium problems arising in mathematical economics typically relies on Knaster-Kuratowski-Mazurkiewicz theorem, Sperner's lemma or some extension involving mapping degree. We propose a different and relatively novel approach where the emphasis is on configuration spaces and equivariant topology, originally developed for applications in discrete and computational geometry (Tverberg type problems, necklace splitting problem in the sense of N.Alon and D. West, etc.). We illustrate the method by proving several relatives (extensions) of the classical envy-free division theorem of David Gale, where the emphasis is on preferences allowing the players to choose degenerate pieces of the cake.
arXiv (Cornell University), Nov 18, 2009
We give a simpler, degree-theoretic proof of the striking new Tverberg type theorem of Blagojević... more We give a simpler, degree-theoretic proof of the striking new Tverberg type theorem of Blagojević, Ziegler and Matschke, arXiv:0910.4987v2. Our method also yields some new examples of "constrained Tverberg theorems" including a simple colored Radon's theorem for d + 3 points in R d. This gives us an opportunity to review some of the highlights of this beautiful theory and reexamine the role of chessboard complexes in these and related problems of topological combinatorics. * This paper is an expanded version of our preprint [X], with added Theorem 4 and its consequences. * Supported by Grants 144014 and 144026 of the Serbian Ministry of Science and Technology. * After the preliminary version [X] of our paper was released and shared with a circle of specialists, we were kindly informed by P. Blagojević that B. Matschke has also discovered a proof that simplifies their original approach. This proof is incorporated in [BMZ-2].
Journal of Combinatorial Theory, Series A, 2017
Following D.B. Karaguezian, V. Reiner, and M.L. Wachs (Matching Complexes, Bounded Degree Graph C... more Following D.B. Karaguezian, V. Reiner, and M.L. Wachs (Matching Complexes, Bounded Degree Graph Complexes, and Weight Spaces of GL-Complexes, Journal of Algebra 2001) we study the connectivity degree and shellability of multiple chessboard complexes. Our central new results (Theorems 3.2 and 4.4) provide sharp connectivity bounds relevant to applications in Tverberg type problems where multiple points of the same color are permitted. These results also provide a foundational work for the new results of Tverberg-van Kampen-Flores type, as announced in the forthcoming paper [JVZ-2].
Advances in Mathematics, Jun 1, 2008
The well-known "splitting necklace theorem" of Alon [1] says that each necklace with k • a i bead... more The well-known "splitting necklace theorem" of Alon [1] says that each necklace with k • a i beads of color i = 1,. .. , n can be fairly divided between k "thieves" by at most n(k − 1) cuts. Alon deduced this result from the fact that such a division is possible also in the case of a continuous necklace [0, 1] where beads of given color are interpreted as measurable sets A i ⊂ [0, 1] (or more generally as continuous measures µ i). We demonstrate that Alon's result is a special case of a multidimensional, consensus division theorem of n continuous probability measures µ 1 ,. .. , µn on a d-cube [0, 1] d. The dissection is performed by m 1 +. .. + m d = n(k − 1) hyperplanes parallel to the sides of [0, 1] d dividing the cube into m 1 •. .. • m d elementary parallelepipeds where the integers m i are prescribed in advance.
arXiv (Cornell University), Jul 25, 2016
We introduce and study Alexander r-tuples K = K i r i=1 of simplicial complexes, as a common gene... more We introduce and study Alexander r-tuples K = K i r i=1 of simplicial complexes, as a common generalization of pairs of Alexander dual complexes (Alexander 2-tuples) and r-unavoidable complexes of [BFZ-1]. In the same vein, the Bier complexes, defined as the deleted joins K * ∆ of Alexander r-tuples, include both standard Bier spheres and optimal multiple chessboard complexes (Section 2.2) as interesting, special cases. Our main results are Theorem 4.3 saying that (1) the r-fold deleted join of Alexander r-tuple is a pure complex homotopy equivalent to a wedge of spheres, and (2) the r-fold deleted join of a collective unavoidable r-tuple is (n − r − 1)-connected, and a classification theorem (Theorem 5.1 and Corollary 5.2) for Alexander r-tuples and Bier complexes.
arXiv (Cornell University), Aug 1, 2021
A Bier sphere Bier(K) = K * ∆ K • , defined as the deleted join of a simplicial complex and its A... more A Bier sphere Bier(K) = K * ∆ K • , defined as the deleted join of a simplicial complex and its Alexander dual K • , is a purely combinatorial object (abstract simplicial complex). Here we study a hidden geometry of Bier spheres by describing their natural geometric realizations, compute their volume, describe an effective criterion for their polytopality, and associate to K a natural fan F an(K), related to the Braid fan. Along the way we establish a connection of Bier spheres of maximal volume with recent generalizations of the classical Van Kampen-Flores theorem and clarify the role of Bier spheres in the theory of generalized permutohedra.
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Papers by Rade Zivaljevic