arXiv subjects

Rade T. Živaljević

Publications and source records attributed to Rade T. Živaljević.

18 recordsLinked to original sources

arXiv2025

Indecomposability of the median hypersimplex and polytopality of the hemi-icosahedral Bier sphere

We prove that the median hypersimplex $Δ_{2k,k}$ is Minkowski indecomposable, i.e. it cannot be expressed as a non-trivial Minkowski sum $Δ_{2k,k} = P+Q$, where $P\neq λΔ_{2k,k}\neq Q$. We obtain as a corollary that $Δ_{2k,k}$ represents a ray in the submodular cone (the deformation cone of the permutahedron). Building on the previously developed geometric methods and extensive computer search, we exhibit a twelve vertex, $4$-dimensional polytopal realization of the Bier sphere of the hemi-icosahedron, the vertex minimal triangulation of the real projective plane.

math.CO↗

arXiv2023

Polytopality of simple games

The Bier sphere $Bier(\mathcal{G}) = Bier(K) = K\ast_ΔK^\circ$ and the canonical fan $Fan(Γ) = Fan(K)$ are combinatorial/geometric companions of a simple game $\mathcal{G} = (P,Γ)$ (equivalently the associated simplicial complex $K$), where $P$ is the set of players, $Γ\subseteq 2^P$ is the set of wining coalitions, and $K = 2^P\setminus Γ$ is the simplicial complex of losing coalitions. We characterize roughly weighted majority games as the games $Γ$ such that $Bier(\mathcal{G})$ (respectively $Fan(Γ)$) is canonically polytopal (canonically pseudo-polytopal) and show, by an experimental/theoretical argument, that all simple games with at most five players are polytopal.

math.CO↗

arXiv2022

Colored Tverberg Theorems for non-prime powers

We prove a relative of the Optimal (Type B)} Colored Tverberg theorem of Živaljević and Vrećica which modifies this results in two different ways. (1) Our result is valid if the number of rainbow faces is $q= p^n-1$, where $p$ is a prime. (2) The size of rainbow simplices satisfies the condition $\vert C_i\vert \in \{2q-2, 2q+1\}$ while in the original theorem $\vert C_i\vert = 2q-1$ for all $i$.

math.CO↗

arXiv2021

Bier spheres of extremal volume and generalized permutohedra

A Bier sphere $Bier(K) = K\ast_ΔK^\circ$, defined as the deleted join of a simplicial complex and its Alexander dual $K^\circ$, 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 $Fan(K)$, 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.

math.CO↗

arXiv2021

Optimal colored Tverberg theorems for prime powers

The type A colored Tverberg theorem of Blagojević, Matschke, and Ziegler provides optimal bounds for the colored Tverberg problem, under the condition that the number of intersecting rainbow simplices is a prime number. We extend this result to an optimal, type A colored Tverberg theorem for multisets of colored points, which is valid for each prime power $r=p^k$. One of the principal new ideas is to replace the ambient simplex $Δ^N$, used in the original Tverberg theorem, by an "abridged simplex" of smaller dimension, and to compensate for this reduction by allowing vertices to repeatedly appear a controlled number of times in different rainbow simplices. Configuration spaces, used in the proof, are combinatorial pseudomanifolds which can be represented as multiple chessboard complexes. Our main topological tool is the Eilenberg-Krasnoselskii theory of degrees of equivariant maps for non-free actions.

math.MG↗

arXiv2020

Generalized Tonnetz and discrete Abel-Jacobi map

Motivated by classical Euler's $Tonnetz$, we introduce and study the combinatorics and topology of more general simplicial complexes $Tonn^{n,k}(L)$ of "Tonnetz type". Out main result is that for a sufficiently generic choice of parameters the generalized tonnetz $Tonn^{n,k}(L)$ is a triangulation of a $(k-1)$-dimensional torus $T^{k-1}$. In the proof we construct and use the properties of a "discrete Abel-Jacobi map", which takes values in the torus $T^{k-1} \cong \mathbb{R}^{k-1}/Λ$ where $Λ\cong \mathbb{A}^\ast_{k-1}$ is the permutohedral lattice.

math.MG↗

arXiv2020

Generalized chessboard complexes and discrete Morse theory

Chessboard complexes and their generalizations, as objects, and Discrete Morse theory, as a tool, are presented as a unifying theme linking different areas of geometry, topology, algebra and combinatorics. Edmonds and Fulkerson bottleneck (minmax) theorem is proved and interpreted as a result about a critical point of a discrete Morse function on the Bier sphere of an associated simplicial complex $K$. We illustrate the use of "standard discrete Morse functions" on generalized chessboard complexes by proving a connectivity result for chessboard complexes with multiplicities. Applications include new Tverberg-Van Kampen-Flores type results for $j$-wise disjoint partitions of a simplex.

math.MG↗

arXiv2020

Colored Tverberg problem, extensions and new results

We prove a "multiple colored Tverberg theorem" and a "balanced colored Tverberg theorem", by applying different methods, tools and ideas. The proof of the first theorem uses multiple chessboard complexes (as configuration spaces) and Eilenberg-Krasnoselskii theory of degrees of equivariant maps for non-free actions. The proof of the second result relies on high connectivity of the configuration space, established by discrete Morse theory.

math.MG↗

arXiv2019

Polytopal Bier spheres and Kantorovich-Rubinstein polytopes of weighted cycles

The problem of deciding if a given triangulation of a sphere can be realized as the boundary sphere of a simplicial, convex polytope is known as the "Simplicial Steinitz problem". It is known by an indirect and non-constructive argument that a vast majority of Bier spheres are non-polytopal. Contrary to that, we demonstrate that the Bier spheres associated to threshold simplicial complexes are all polytopal. Moreover, we show that all Bier spheres are starshaped. We also establish a connection between Bier spheres and Kantorovich-Rubinstein polytopes by showing that the boundary sphere of the KR-polytope associated to a polygonal linkage (weighted cycle) is isomorphic to the Bier sphere of the associated simplicial complex of "short sets".

math.MG↗

arXiv2019

Combinatorics of `unavoidable complexes'

The partition number $π(K)$ of a simplicial complex $K\subset 2^{[n]}$ is the minimum integer $ν$ such that for each partition $A_1\uplus\ldots\uplus A_ν= [n]$ of $[n]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is $r$-unavoidable if $π(K)\leq r$. Motivated by the problems of Tverberg-Van Kampen-Flores type, and inspired by the `constraint method' of Blagojević, Frick, and Ziegler, arXiv:1401.0690 [math.CO], we study the combinatorics of $r$-unavoidable complexes.

math.CO↗

arXiv2018

Topology of unavoidable complexes

The partition number $π(K)$ of a simplicial complex $K\subset 2^{[m]}$ is the minimum integer $ν$ such that for each partition $A_1\uplus\ldots\uplus A_ν= [m]$ of $[m]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is $r$-unavoidable if $π(K)\leq r$. We say that a complex $K$ is globally $r$-non-embeddable in $\mathbb{R}^d$ if for each continuous map $f: | K| \rightarrow \mathbb{R}^d$ there exist $r$ vertex disjoint faces $σ_1,\ldots, σ_r$ of $| K|$ such that $f(σ_1)\cap\ldots\cap f(σ_r)\neq\emptyset$. Motivated by the problems of Tverberg-Van Kampen-Flores type we prove several results (Theorems 3.6, 3.9, 4.6) which link together the combinatorics and topology of these two classes of complexes. One of our central observations (Theorem 4.6), summarizing and extending results of G. Schild, B. Grünbaum and many others, is that interesting examples of (globally) $r$-non-embeddable complexes can be found among the joins $K = K_1\ast\ldots\ast K_s$ of $r$-unavoidable complexes.

math.AT↗

arXiv2018

Cyclohedron and Kantorovich-Rubinstein polytopes

We show that the cyclohedron (Bott-Taubes polytope) $W_n$ arises as the dual of a Kantorovich-Rubinstein polytope $KR(ρ)$, where $ρ$ is a quasi-metric (asymmetric distance function) satisfying strict triangle inequality. From a broader perspective, this phenomenon illustrates the relationship between a nestohedron $Δ_{\mathcal{\widehat{F}}}$ (associated to a building set $\mathcal{\widehat{F}}$) and its non-simple deformation $Δ_{\mathcal{F}}$, where $\mathcal{F}$ is an `irredundant' or `tight basis' of $\mathcal{\widehat{F}}$. Among the consequences are a new proof of a recent result of Gordon and Petrov (arXiv:1608.06848 [math.CO]) 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.

math.CO↗

arXiv2016

A glimpse into continuous combinatorics of posets, polytopes, and matroids

This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and other combinatorial structures. Among the illustrative examples are an Euler formula for a class of `continuous convex polytopes' (conjectured by Kalai and Wigderson), a duality result for a class of `continuous matroids', a calculation of the Euler characteristic of ideals in the Grassmannian poset (related to a problem of Gian-Carlo Rota), an exposition of the `homotopy complementation formula' for topological posets and its relation to the results of Kallel and Karoui about `weighted barycenter spaces' and a conjecture of Vassiliev about simplicial resolutions of singularities. We also include an extension of the index inequality (Sarkaria's inequality) based on interpreting diagrams of spaces as continuous posets.

math.CO↗

arXiv2016

Hyperplane mass equipartition problem and the shielding functions of Ramos

We give a proof of the result of Edgar Ramos which claims that two finite, continuous Borel measures $μ_1$ and $μ_2$ defined on $\mathbb{R}^5$ admit an equipartition by a collection of three hyperplanes. Our proof illuminates one of the central methods developed and used in our earlier papers and may serve as a good `test case' for addressing (and resolving) the `issues' raised in the paper "Topology of the Grünbaum-Hadwiger-Ramos hyperplane mass partition problem", arXiv:1502.02975 [math.AT]. We also offer a degree-theoretic interpretation of the `parity calculation method' developed by Ramos and demonstrate that, up to minor corrections or modifications, it remains a rigorous and powerful tool for proving results about mass equipartitions.

math.MG↗

arXiv2015

Multiple chessboard complexes and the colored Tverberg problem

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 foundation for the new results of Tverberg-van Kampen-Flores type, as announced in arXiv:1502.05290 [math.CO].

math.CO↗

arXiv2014

Signed polyomino tilings by n-in-line polyominoes and Groebner bases

Conway and Lagarias observed that a triangular region T(m) in a hexagonal lattice admits signed tiling by three-in-line polyominoes (tribones) if and only if m=9d-1 or m=9d for some integer d. We apply the theory of Groebner bases over integers to show that T(m) admits a signed tiling by n-in-line polyominoes (n-bones) if and only if m=dn^2-1 or m=dn^2 for some integer d. Explicit description of the Groebner basis allows us to calculate the "Groebner discrete volume" of a lattice region by applying the division algorithm to its `Newton polynomial'. Among immediate consequences is a description of the tile homology group of the $n$-in-line polyomino.

math.CO↗

arXiv2013

Illumination complexes, Δ-zonotopes, and the polyhedral curtain theorem

Illumination complexes are examples of 'flat polyhedral complexes' which arise if several copies of a convex polyhedron (convex body) Q are glued together along some of their common faces (closed convex subsets of their boundaries). A particularly nice example arises if Q is a Δ-zonotope (generalized rhombic dodecahedron), known also as the dual of the difference body Δ - Δ of a simplex Δ, or the dual of the convex hull of the root system A_n. We demonstrate that the illumination complexes and their relatives can be used as 'configuration spaces', leading to new 'fair division theorems'. Among the central new results is the 'polyhedral curtain theorem' (Theorem 3) which is a relative of both the 'ham sandwich theorem' and the 'splitting necklaces theorem'.

math.MG↗

arXiv1998

Combinatorics of Topological Posets:\ Homotopy complementation formulas

We show that the well known {\em homotopy complementation formula} of Björner and Walker admits several closely related generalizations on different classes of topological posets (lattices). The utility of this technique is demonstrated on some classes of topological posets including the Grassmannian and configuration posets, $\widetilde{\mathbf G}_n(R)$ and $\exp_n(X)$ which were introduced and studied by V.~Vassiliev. Among other applications we present a reasonably complete description, in terms of more standard spaces, of homology types of configuration posets $\exp_n(S^m)$ which leads to a negative answer to a question of Vassilev raised at the workshop ``Geometric Combinatorics'' (MSRI, February 1997).

math.CO↗