5 papers · 1 filter
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…