arXiv subjects

Pavle V. M. Blagojević

Publications and source records attributed to Pavle V. M. Blagojević.

At least 19 recordsLinked to original sources

arXiv2026

Cone and constrained colorful Carathéodory Theorems

Holmsen proved in 2016 a generalization of the classical colorful Caratheodory theorem in which a matroid imposes additional constraints on the desired colorful transversal. His approach also works in the more general setting of oriented matroids, rather than relying directly on convex hulls. In this paper, we extend these ideas in several directions. First, we study which colorful Caratheodory-type results remain valid when convex cones replace convex hulls, as well as analogous modifications in the oriented matroid setting. Second, we consider variants in which the additional constraint on the transversal is not encoded by a matroid. This leads to new extensions of the classical Tverberg theorem. Our approach is topological, following the methods of Holmsen, and Kalai and Meshulam, on which it builds. The key idea is to analyze homology groups of simplicial complexes that encode colorful Caratheodory-type phenomena, such as the support complex of an oriented matroid. In particular, one shows that these complexes are (near-)d-Leray. We extend this analysis by carrying out more detailed homology computations for these complexes, with the aim of enabling further and more refined applications of the method.

math.CO↗

arXiv2022

Topology of the Grünbaum--Hadwiger--Ramos problem for mass assignments

In this paper, motivated by recent work of Schnider and Axelrod-Freed \& Soberón, we study an extension of the classical Grünbaum--Hadwiger--Ramos mass partition problem to mass assignments. Using the Fadell--Husseini index theory we prove that for a given family of $j$ mass assignments $μ_1,\dots,μ_j$ on the Grassmann manifold $G_{\ell}(\R^d)$ and a given integer $k\geq 1$ there exist a linear subspace $L\in G_{\ell}(\R^d)$ and $k$ affine hyperplanes in $L$ that equipart the masses $μ_1^L,\dots,μ_j^L$ assigned to the subspace $L$, provided that $d\geq j + (2^{k-1}-1)2^{\lfloor\log_2j\rfloor}$.

math.AT↗

arXiv2021

Facet volumes of polytopes

In this paper, motivated by the work of Edelman and Strang, we show that for fixed integers $d\geq 2$ and $n\geq d+1$ the configuration space of all facet volume vectors of all $d$-polytopes in $\mathbb R^{d}$ with $n$ facets is a full dimensional cone in $\mathbb R^{n}$. In particular, for tetrahedra ($d=3$ and $n=4$) this is a cone over a regular octahedron. Our proof is based on a novel configuration space / test map scheme which uses topological methods for finding solutions of a problem, and tools of differential geometry to identify solutions with the desired properties. Furthermore, our results open a possibility for the study of realization spaces of all $d$-polytopes in $\mathbb R^{d}$ with $n$ facets by the methods of algebraic topology.

math.CO↗

arXiv2021

Equivariant Cohomology of Configuration Spaces mod 2: The State of the Art

The equivariant cohomology of the classical configuration space $F(\mathbb{R}^d,n)$ has been been of great interest and has been studied intensively starting with the classical papers by Artin (1925/1947) on the theory of braids, by Fox and Neuwirth (1962), Fadell and Neuwirth (1962), and Arnol'd (1969). We give a brief treatment of the subject from the beginnings to recent developments. However, we focus on the mod 2 equivariant cohomology algebras of the classical configuration space $F(\mathbb{R}^d,n)$, as described in an influential paper by Hung (1990). We show with a new, detailed proof that his main result is correct, but that the arguments that were given by Hung on the way to his result are not, as are some of the intermediate results in his paper. This invalidates a paper by three of the present authors, Blagojević, Lück \& Ziegler (2016), who used a claimed intermediate result from Hung (1990) in order to derive lower bounds for the existence of $k$-regular and $\ell$-skew embeddings. Using our new proof for Hung's main result, we get new lower bounds for existence of highly regular embeddings: Some of them agree with the previously claimed bounds, some are weaker.

math.AT↗

arXiv2021

More bisections by hyperplane arrangements

A union of an arrangement of affine hyperplanes $H$ in $R^d$ is the real algebraic variety associated to the principal ideal generated by the polynomial $p_{H}$ given as the product of the degree one polynomials which define the hyperplanes of the arrangement. A finite Borel measure on $R^d$ is bisected by the arrangement of affine hyperplanes $H$ if the measure on the "non-negative side" of the arrangement $\{x\in R^d : p_{H}(x)\ge 0\}$ is the same as the measure on the "non-positive" side $\{x\in R^d : p_{H}(x)\le 0\}$. In 2017 Barba, Pilz \& Schnider considered special cases of the following measure partition hypothesis: For a given collection of $j$ finite Borel measures on $R^d$ there exists a $k$-element affine hyperplane arrangement that bisects each of the measures into equal halves simultaneously. They showed that there are simultaneous bisections in the case when $d=k=2$ and $j=4$. They conjectured that every collection of $j$ measures on $R^d$ can be simultaneously bisected with a $k$-element affine hyperplane arrangement provided that $d\ge \lceil j/k \rceil$. The conjecture was confirmed in the case when $d\ge j/k=2^a$ by Hubard and Karasev in 2018. In this paper we give a different proof of the Hubard and Karasev result using the framework of Blagojević, Frick, Haase \& Ziegler (2016), based on the equivariant relative obstruction theory of tom Dieck, which was developed for handling the Grünbaum--Hadwiger--Ramos hyperplane measure partition problem. Furthermore, this approach allowed us to prove even more, that for every collection of $2^a(2h+1)+\ell$ measures on $R^{2^a+\ell}$, where $1\leq \ell\leq 2^a-1$, there exists a $(2h+1)$-element affine hyperplane arrangement that bisects all of them simultaneously. Our result was extended to the case of spherical arrangements and reproved by alternative methods in a beautiful way by Crabb in 2020.

math.MG↗

arXiv2020

Counting Periodic Trajectories of Finsler Billiards

We provide lower bounds on the number of periodic Finsler billiard trajectories inside a quadratically convex smooth closed hypersurface $M$ in a $d$-dimensional Finsler space with possibly irreversible Finsler metric. An example of such a system is a billiard in a sufficiently weak magnetic field. The $r$-periodic Finsler billiard trajectories correspond to $r$-gons inscribed in $M$ and having extremal Finsler length. The cyclic group ${\mathbb Z}_r$ acts on these extremal polygons, and one counts the ${\mathbb Z}_r$-orbits. Using Morse and Lusternik-Schnirelmann theories, we prove that if $r\ge 3$ is prime, then the number of $r$-periodic Finsler billiard trajectories is not less than $(r-1)(d-2)+1$. We also give stronger lower bounds when $M$ is in general position. The problem of estimating the number of periodic billiard trajectories from below goes back to Birkhoff. Our work extends to the Finsler setting the results previously obtained for Euclidean billiards by Babenko, Farber, Tabachnikov, and Karasev.

math.DS↗

arXiv2019

Cutting a part from many measures

Holmsen, Kynčl and Valculescu recently conjectured that if a finite set $X$ with $\ell n$ points in $\mathbb{R}^d$ that is colored by $m$ different colors can be partitioned into $n$ subsets of $\ell$ points each, such that each subset contains points of at least $d$ different colors, then there exists such a partition of $X$ with the additional property that the convex hulls of the $n$ subsets are pairwise disjoint. We prove a continuous analogue of this conjecture, generalized so that each subset contains points of at least $c$ different colors, where we also allow $c$ to be greater than $d$. Furthermore, we give lower bounds on the fraction of the points each of the subsets contains from $c$ different colors. For example, when $n\geq 2$, $d\geq 2$, $c\geq d$ with $m\geq n(c-d)+d$ are integers, and $μ_1, \dots, μ_m$ are $m$ positive finite absolutely continuous measures on $\mathbb{R}^d$, we prove that there exists a partition of $\mathbb{R}^d$ into $n$ convex pieces which equiparts the measures $μ_1, \dots, μ_{d-1}$, and in addition every piece of the partition has positive measure with respect to at least $c$ of the measures $μ_1, \dots, μ_m$.

math.CO↗

arXiv2019

Barycenters of Polytope Skeleta and Counterexamples to the Topological Tverberg Conjecture, via Constraints

Using the authors' 2014 "constraints method," we give a short proof for a 2015 result of Dobbins on representations of a point in a polytope as the barycenter of points in a skeleton, and show that the "r-fold Whitney trick" of Mabillard and Wagner (2014/2015) implies that the Topological Tverberg Conjecture for r-fold intersections fails dramatically for all r that are not prime powers.

math.CO↗

arXiv2019

Some more amplituhedra are contractible

The amplituhedra arise as images of the totally nonnegative Grassmannians by projections that are induced by linear maps. They were introduced in Physics by Arkani-Hamed \& Trnka (Journal of High Energy Physics, 2014) as model spaces that should provide a better understanding of the scattering amplitudes of quantum field theories. The topology of the amplituhedra has been known only in a few special cases, where they turned out to be homeomorphic to balls. The amplituhedra are special cases of Grassmann polytopes introduced by Lam (Current Developments in Mathematics 2014, Int.\ Press). In this paper we show that that some further amplituhedra are homeomorphic to balls, and that some more Grassmann polytopes and amplituhedra are contractible.

math.GT↗

arXiv2018

Orthogonal shadows and index of Grassmann manifolds

In this paper we study the $\Z/2$ action on real Grassmann manifolds $G_{n}(\R^{2n})$ and $\widetilde{G}_{n}(\R^{2n})$ given by taking (appropriately oriented) orthogonal complement. We completely evaluate the related $\Z/2$ Fadell--Husseini index utilizing a novel computation of the Stiefel--Whitney classes of the wreath product of a vector bundle. These results are used to establish the following geometric result about the orthogonal shadows of a convex body: For $n=2^a (2 b+1)$, $k=2^{a+1}-1$, $C$ a convex body in $\R^{2n}$, and $k$ real valued functions $α_1,\ldots,α_k$ continuous on convex bodies in $\R^{2n}$ with respect to the Hausdorff metric, there exists a subspace $V\subseteq\R^{2n}$ such that projections of $C$ to $V$ and its orthogonal complement $V^{\perp}$ have the same value with respect to each function $α_i$, which is $α_i (p_V(C))=α_i (p_{V^\perp} (C))$ for all $1\leq i\leq k$.

math.AT↗

arXiv2018

A center transversal theorem for an improved Rado depth

A celebrated result of Dol'nikov, and of Živaljević and Vrećica, asserts that for every collection of $m$ measures $μ_1,\dots,μ_m$ on the Euclidean space $\mathbb R^{n + m - 1}$ there exists a projection onto an $n$-dimensional vector subspace $Γ$ with a point in it at depth at least $\tfrac{1}{n + 1}$ with respect to each associated $n$-dimensional marginal measure $Γ_*μ_1,\dots,Γ_*μ_m$. In this paper we consider a natural extension of this result and ask for a minimal dimension of a Euclidean space in which one can require that for any collection of $m$ measures there exists a vector subspace $Γ$ with a point in it at depth slightly greater than $\tfrac{1}{n + 1}$ with respect to each $n$-dimensional marginal measure. In particular, we prove that if the required depth is $\tfrac{1}{n + 1} + \tfrac{1}{3(n + 1)^3}$ then the increase in the dimension of the ambient space is a linear function in both $m$ and $n$.

math.MG↗

arXiv2017

Convex equipartitions of colored point sets

We show that any $d$-colored set of points in general position in $\mathbb{R}^d$ can be partitioned into $n$ subsets with disjoint convex hulls such that the set of points and all color classes are partitioned as evenly as possible. This extends results by Holmsen, Kynčl & Valculescu (2017) and establishes a special case of their general conjecture. Our proof utilizes a result obtained independently by Soberón and by Karasev in 2010, on simultaneous equipartitions of $d$ continuous measures in $\mathbb{R}^d$ by $n$ convex regions. This gives a convex partition of $\mathbb{R}^d$ with the desired properties, except that points may lie on the boundaries of the regions. In order to resolve the ambiguous assignment of these points, we set up a network flow problem. The equipartition of the continuous measures gives a fractional flow. The existence of an integer flow then yields the desired partition of the point set.

math.CO↗

arXiv2017

Thieves can make sandwiches

We prove a common generalization of the Ham Sandwich theorem and Alon's Necklace Splitting theorem. Our main results show the existence of fair distributions of $m$ measures in $R^d$ among $r$ thieves using roughly $mr/d$ convex pieces, even in the cases when $m$ is larger than the dimension. The main proof relies on a construction of a geometric realization of the topological join of two spaces of partitions of $R^d$ into convex parts, and the computation of the Fadell-Husseini ideal valued index of the resulting spaces.

math.CO↗

arXiv2017

Beyond the Borsuk-Ulam theorem: The topological Tverberg story

Bárány's "topological Tverberg conjecture" from 1976 states that any continuous map of an $N$-simplex $Δ_N$ to $\mathbb{R}^d$, for $N\ge(d+1)(r-1)$, maps points from $r$ disjoint faces in $Δ_N$ to the same point in $\mathbb{R}^d$. The proof of this result for the case when $r$ is a prime, as well as some colored version of the same result, using the results of Borsuk-Ulam and Dold on the non-existence of equivariant maps between spaces with a free group action, were main topics of Matoušek's 2003 book "Using the Borsuk-Ulam theorem." In this paper we show how advanced equivariant topology methods allow one to go beyond the prime case of the topological Tverberg conjecture. First we explain in detail how equivariant cohomology tools (employing the Borel construction, comparison of Serre spectral sequences, Fadell-Husseini index, etc.) can be used to prove the topological Tverberg conjecture whenever $r$ is a prime power. Our presentation includes a number of improved proofs as well as new results, such as a complete determination of the Fadell-Husseini index of chessboard complexes in the prime case. Then we introduce the "constraint method," which applied to suitable "unavoidable complexes" yields a great variety of variations and corollaries to the topological Tverberg theorem, such as the "colored" and the "dimension-restricted" (Van Kampen-Flores type) versions. Both parts have provided crucial components to the recent spectacular counter-examples in high dimensions for the case when $r$ is not a prime power.

math.CO↗

arXiv2017

Tverberg-type theorems for matroids: A counterexample and a proof

Bárány, Kalai, and Meshulam recently obtained a topological Tverberg-type theorem for matroids, which guarantees multiple coincidences for continuous maps from a matroid complex to d-dimensional Euclidean space, if the matroid has sufficiently many disjoint bases. They make a conjecture on the connectivity of k-fold deleted joins of a matroid with many disjoint bases, which would yield a much tighter result - but we provide a counterexample already for the case of k=2, where a tight Tverberg-type theorem would be a topological Radon theorem for matroids. Nevertheless, we prove the topological Radon theorem for the counterexample family of matroids by an index calculation, despite the failure of the connectivity-based approach.

math.CO↗

arXiv2017

The topological transversal Tverberg theorem plus constraints

In this paper we use the strength of the constraint method in combination with a generalized Borsuk-Ulam type theorem and a cohomological intersection lemma to show how one can obtain many new topological transversal theorems of Tverberg type. In particular, we derive a topological generalized transversal Van Kampen-Flores theorem and a topological transversal weak colored Tverberg theorem.

math.CO↗

arXiv2016

Hyperplane mass partitions via relative equivariant obstruction theory

The Grünbaum-Hadwiger-Ramos hyperplane mass partition problem was introduced by Grünbaum (1960) in a special case and in general form by Ramos (1996). It asks for the "admissible" triples $(d,j,k)$ such that for any $j$ masses in $\mathbb{R}^d$ there are $k$ hyperplanes that cut each of the masses into $2^k$ equal parts. Ramos' conjecture is that the Avis-Ramos necessary lower bound condition $dk\ge j(2^k-1)$ is also sufficient. We develop a "join scheme" for this problem, such that non-existence of an $G_k$-equivariant map between spheres $(S^d)^{*k} \rightarrow S(W_k\oplus U_k^{\oplus j})$ that extends a test map on the subspace of $(S^d)^{*k}$ where the hyperoctahedral group $G_k$ acts non-freely, implies that $(d,j,k)$ is admissible. For the sphere $(S^d)^{*k}$ we obtain a very efficient regular cell decomposition, whose cells get a combinatorial interpretation with respect to measures on a modified moment curve. This allows us to apply relative equivariant obstruction theory successfully, even in the case when the difference of dimensions of the spheres $(S^d)^{*k}$ and $S(W_k\oplus U_k^{\oplus j})$ is greater than one. The evaluation of obstruction classes leads to counting problems for concatenated Gray codes. Thus we give a rigorous, unified treatment of the previously announced cases of the Grünbaum-Hadwiger-Ramos problem, as well as a number of new cases for Ramos' conjecture.

math.AT↗

arXiv2016

Polynomial partitioning for several sets of varieties

We give a new, systematic proof for a recent result of Larry Guth and thus also extend the result to a setting with several families of varieties: For any integer $D\geq 1$ and any collection of sets $Γ_1,\ldots,Γ_j$ of low-degree $k$-dimensional varieties in $\mathbb{R}^n$ there exists a non-zero polynomial $p\in\mathbb{R}[X_1,\ldots,X_n]$ of degree at most $D$ so that each connected component of $\mathbb{R}^n{\setminus}Z(p)$ intersects $O(jD^{k-n}|Γ_i|)$ varieties of $Γ_i$, simultaneously for every $1\leq i\leq j$. For $j=1$ we recover the original result by Guth. Our proof, via an index calculation in equivariant cohomology, shows how the degrees of the polynomials used for partitioning are dictated by the topology, namely by the Euler class being given in terms of a top Dickson polynomial.

math.AT↗