9 papers
KKM meets colorful Carathéodory
Alexander Polyanskii
The colorful Carathéodory theorem states that if satisfy for every , then there exist points such tha…
A short proof of Ball's plank theorem
Arseniy Akopyan, Alexander Polyanskii
We give a short proof of the following theorem of K. Ball: If a centrally symmetric convex body is covered by finitely many planks, then the sum of their rel…
Spanning \(k\)-trees and the colorful Carathéodory theorem
Mikhail Bludov, Alexander Polyanskii
Very recently, using Meshulam's lemma, Blagojević proved a constrained version of the colorful Carathéodory theorem for joins of bipartite spanning trees and wedge of spheres. Our…
Optimal partial plank coverings
Egor Bakaev, Alexander Polyanskii
A plank of width in a Euclidean space is the set of points lying between two parallel hyperplanes at distance from each other. Bang's theorem says that if a family of plank…
Triangle covering problems and the Viterbo inequality in the plane
Alexey Balitskiy, Ivan Mitrofanov, Alexander Polyanskii
We review a certain problem on covering triangles in the plane. Equivalently, it can be viewed as a family of 'isobilliard' inequalities in convex shapes, and as a special case of…
New Helly-type results for discrete boxes: Quantitative colorful and -variants
Rahul Gangopadhyay, Alexander Polyanskii, Wei Rao
In 2008, Halman showed that for any finite set and any finite family of axis-parallel boxes in , if the intersection of and a…
No-dimensional Tverberg-type problems
Alexander Polyanskii
Recently, Adiprasito et al. have initiated the study of the so-called no-dimensional Tverberg problem. This problem can be informally stated as follows: Given , partition…
Tight colorful no-dimensional Tverberg theorem
Polina Barabanshchikova, Grigory Ivanov, Alexander Polyanskii
We study colorful no-dimensional Tverberg-type problems and obtain several optimal results. A colorful no-dimensional Tverberg-type theorem provides a bound on a radius such th…
Alon-Boppana-type bounds for weighted graphs
Alexandr Polyanskii, Rinat Sadykov
The unraveled ball of radius centered at a vertex in a weighted graph is the ball of radius centered at in the universal cover of . We present a general boun…