arXiv subjects

Pavle V. M. Blagojevic

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

11 recordsLinked to original sources

arXiv2025

Colourful Carathéodory's Theorem plus a constraint

We develop a topological framework in an attempt to generalize the classical colourful Caratheodory theorem by imposing an additional constraint. For that we introduce the notion of zero-avoding complexes and covering criteria for the existence of colourful transversals. Using the developed method in combination with the homological Nerve theorem of Meshulam we recover all known versions of the colourful Caratheodory's theorem and prove a constraint extension which, in particular, implies an extension of the original (affine) Tverberg result.

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arXiv2024

Convex Equipartitions inspired by the little cubes operad

A decade ago two groups of authors, Karasev, Hubard and Aronov, and Blagojević and Ziegler, have shown that the regular convex partitions of a Euclidean space into $n$ parts yield a solution to the generalised Nandakumar and Ramana-Rao conjecture when $n$ is a prime power. This was obtained by parametrising the space of regular equipartitions of a given convex body with the classical configuration space. Now, we repeat the process of regular convex equipartitions many times, first partitioning the Euclidean space into $n_1$ parts, then each part into $n_2$ parts, and so on. In this way we obtain iterated convex equipartions of a given convex body into $n=n_1...n_k$ parts. Such iterated partitions are parametrised by the (wreath) product of classical configuration spaces. We develop a new configuration space -- test map scheme for solving the generalised Nandakumar \& Ramana-Rao conjecture using the Hausdorff metric on the space of iterated convex equipartions. The new scheme yields a solution to the conjecture if and only if all the $n_i$'s are powers of the same prime. In particular, for the failure of the scheme outside prime power case we give three different proofs.

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arXiv2024

Many partitions of mass assignments

In this paper, extending the recent work of authors with Calles Loperena and Dimitrijević Blagojević, we give a general and complete treatment of a problem of partition of mass assignments with prescribed arrangements of hyperplanes on Euclidean vector bundles. Using a new configuration test map scheme, as well as an alternative topological framework, we are able to reprove known results, extend them to arbitrary bundles as well as to put various types of constraints on the solutions. Moreover, the developed topological methods allow us to give new proofs and extend results of Guth and Katz, Schnider, and Soberón and Takahashi. In this way we place all these results under one ``roof''.

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arXiv2019

Plus minus analogues for affine Tverberg type results

The classical 1966 theorem of Tverberg with its numerous variations was and still is a motivating force behind many important developments in convex and computational geometry as well as the testing ground for methods from equivariant algebraic topology. In 2018, Bárány and Soberón presented a new variation, the "Tverberg plus minus theorem." In this paper, we give a new proof of the Tverberg plus minus theorem, by using a projective transformation. The same tool allows us to derive plus minus analogues of all known affine Tverberg type results. In particular, we prove a plus minus analogue of the optimal colored Tverberg theorem.

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arXiv2018

Topology of the Grünbaum-Hadwiger-Ramos hyperplane mass partition problem

In 1960 Grünbaum asked whether for any finite mass in $\mathbb{R}^d$ there are $d$ hyperplanes that cut it into $2^d$ equal parts. This was proved by Hadwiger (1966) for $d\le3$, but disproved by Avis (1984) for $d\ge5$, while the case $d=4$ remained open. More generally, Ramos (1996) asked for the smallest dimension $Δ(j,k)$ in which for any $j$ masses there are $k$ affine hyperplanes that simultaneously cut each of the masses into $2^k$ equal parts. At present the best lower bounds on $Δ(j,k)$ are provided by Avis (1984) and Ramos (1996), the best upper bounds by Mani-Levitska, Vrećica \& Živaljević (2006). The problem has been an active testing ground for advanced machinery from equivariant topology. We give a critical review of the work on the Grünbaum--Hadwiger--Ramos problem, which includes the documentation of essential gaps in the proofs for some previous claims. Furthermore, we establish that $Δ(j,2)= \frac12(3j+1)$ in the cases when $j-1$ is a power of $2$, $j\ge5$.

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arXiv2014

On highly regular embeddings

Given parameters k, l, and d, we give new lower bounds on the dimensions N such that there are maps from R^d to R^N that are k-regular, l-skew embeddings, or k-regular-l-skew embeddings. This extends and sharpens results due to Chisholm (1979) and Ghomi-Tabachnikov (2008).

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arXiv2008

Tetrahedra on deformed spheres and integral group cohomology

We show that for every injective continuous map f: S^2 --> R^3 there are four distinct points in the image of f such that the convex hull is a tetrahedron with the property that two opposite edges have the same length and the other four edges are also of equal length. This result represents a partial result for the topological Borsuk problem for R^3. Our proof of the geometrical claim, via Fadell-Husseini index theory, provides an instance where arguments based on group cohomology with integer coefficients yield results that cannot be accessed using only field coefficients.

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arXiv2008

Borsuk-Ulam Theorems for Complements of Arrangements

In combinatorial problems it is sometimes possible to define a $G$-equivariant mapping from a space $X$ of configurations of a system to a Euclidean space $\mathbb{R}^m$ for which a coincidence of the image of this mapping with an arrangement $\mathcal{A}$ of linear subspaces insures a desired set of linear conditions on a configuration. Borsuk-Ulam type theorems give conditions under which no $G$-equivariant mapping of $X$ to the complement of the arrangement exist. In this paper, precise conditions are presented which lead to such theorems through a spectral sequence argument. We introduce a blow up of an arrangement whose complement has particularly nice cohomology making such arguments possible. Examples are presented that show that these conditions are best possible.

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arXiv2007

Using Equivariant Obstruction Theory in Combinatorial Geometry

A significant group of problems coming from the realm of Combinatorial Geometry can only be approached through the use of Algebraic Topology. From the first such application to Kneser's problem in 1978 by Lov% ász \cite{Lovasz} through the solution of the Lovász conjecture \cite% {Babson-Kozlov}, \cite{Carsten}, many methods from Algebraic Topology have been used. Specifically, it appears that the understanding of equivariant theories is of the most importance. The solution of many problems depends on the existence of an elegantly constructed equivariant map. For example, the following problems were approached by discussing the existence of appropriate equivariant maps.A variety of results from algebraic topology were applied in solving these problems. The methods used ranged from well known theorems like Borsuk Ulam and Dold theorem to the integer / ideal valued index theories. In this paper we are going to extract the essence of the equivariant obstruction theory in order to obtain an effective \emph{general position map%} scheme for analyzing the problem of existence of equivariant maps.

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arXiv2004

Topology of partition of measures by fans and the second obstruction

\noindent The simultaneous partition problems are classical problems of the combinatorial geometry which have the natural flavor of the equivariant topology. The $k$-fan partition problems have attracted a lot of attention \cite{Aki2000}, \cite{BaMa2001}, \cite{BaMa2002} and forced some hard concrete combinatorial calculations in the equivariant cohomology \cite% {Bl-Vr-Ziv}. These problems can be reduced, by a beautiful scheme of \cite% {BaMa2001}, to a \textquotedblright typical\textquotedblright question of the existence of a $\mathbb{D}_{2n}$ equivariant map $f:V_{2}(\mathbb{R}% ^{3})\to W_{n}-\cup \mathcal{A}(α)$, where $V_{2}(\mathbb{R}% ^{3})$ is the space of all orthonormal 2-frames in $\mathbb{R}^{3}$ and $% W_{n}-\cup \mathcal{A}(α)$ is the complement of the appropriate arrangement. We introduce the \textit{target extension scheme} which allow us to use the equivariant obstruction theory as a tool for proving that: for every two proper measures on the sphere $S^{2}$, and any $α=(a,a+b,b)\in \mathbb{R}_{>0}^{3}$, there exists an $α$-partition of theses measures by a 3-fan. \noindent The significance of these results, among other, is that, beside negative results \cite{Bl-Vr-Ziv}, the equivariant obstruction theory can pull off some positive results, which were not attained by other means.

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arXiv2004

Computational Topology of Equivariant Maps from Spheres to Complements of Arrangements

We use a well known problem in discrete and computational geometry (partitions of measures by $k$-fans) as a motivation and as a point of departure to illustrate many aspects, both theoretical and computational, of the problem of calculating the obstructions for the existence of equivariant maps. A variety of techniques are introduced and discussed with the emphasis on concrete and explicit calculations. This eventually leads (Theorems 18 and 19) to an almost exhaustive analysis of when such maps do or do not exist in the particular case of interest.

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