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Alexander Polyanskii

Publications and source records attributed to Alexander Polyanskii.

6 recordsLinked to original sources
arXiv2026

Optimal partial plank coverings

A plank of width $w$ in a Euclidean space is the set of points lying between two parallel hyperplanes at distance $w$ from each other. Bang's theorem says that if a family of planks covers a convex body $K$, then their total width is at least the width of $K$, that is, the width of the thinnest plank containing $K$. We study a quantitative variant of this problem in the case where the total width of the planks is fixed. How should the planks be placed so as to cover as much of the volume of the body as possible? For the central case where $K$ is a Euclidean ball, K\'aroly Bezdek asked whether the optimal arrangement consists of a single plank centered at the origin. We give an affirmative answer to this question. We also show that for every planar convex body an optimal partial covering is attained by a single plank.

math.MG↗
arXiv2026

Spanning \(k\)-trees and the colorful Carath\'eodory theorem

Very recently, using Meshulam's lemma, Blagojevi\'c proved a constrained version of the colorful Carath\'eodory theorem for joins of bipartite spanning trees and wedge of spheres. Our main contribution extends his result from joins of bipartite spanning trees with wedges of spheres to joins of spanning \(k\)-trees with wedges of spheres. Our proof is elementary and avoids the topological machinery. We also discuss a homological variation of spanning \(k\)-trees and some Carath\'eodory-type results for them.

math.CO↗
arXiv2026

Triangle covering problems and the Viterbo inequality in the plane

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 Viterbo's conjecture in symplectic geometry. We give an elementary overview of these topics and, using the optics of the covering problem, we establish several new special cases of Viterbo's conjecture, provide a simple explanation of the counterexample of Haim-Kislev and Ostrover, and state a few open questions. The main novel result is a proof of Viterbo's conjecture for lagrangian products $K \times Q$, where $Q \subset \mathbb{R}^2$ is any quadrilateral and $K \subset \mathbb{R}^2$ is any convex shape.

math.MG↗
arXiv2025

New Helly-type results for discrete boxes: Quantitative colorful and $(p,q)$-variants

In 2008, Halman showed that for any finite set $P\subset \mathbb R^d$ and any finite family $\mathcal{B}$ of axis-parallel boxes in $\mathbb{R}^d$, if the intersection of $P$ and any subfamily $\mathcal{B}' \subseteq\mathcal{B}$ of size at most $2d$ is non-empty, then the intersection of $P$ and $\mathcal{B}$ is also non-empty. Very recently Edwards and Sober\'on initiated the study of quantitative colorful version for $2d$ families, $(p,q)$-type variation for $p\geq q\geq d+1$, and other extensions of this Helly-type result by Halman. In this paper, we study the quantitative colorful Halman problem for $2d-1$ families as well its $(p,q)$-type variation for $p\geq q\geq 2$. Specifically, our main result asserts that for any finite set $P$ and finite families of boxes $\mathcal{B}_1,\dots,\mathcal{B}_{2d-1}$ in $\mathbb R^d$, where $d\geq 2$, if every transversal $\mathcal{B}$ for the families has an intersection $\bigcap \mathcal{B}$ containing at least $n$ points of $P$, then there exist $j\in[2d-1]$ and a subset of $P$ of size at most \[ 2n+\Big\lfloor \frac{n-1}{d \cdot 2^{d-1}} \Big\rfloor, \] such that each box of $\mathcal{B}_j$ contains at least $n$ points of this subset.

math.CO↗
arXiv2025

No-dimensional Tverberg-type problems

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 $n\geq k$, partition an $n$-point set in Euclidean space into $k$ parts such that their convex hulls intersect a ball of relatively small radius. In this survey, we aim to present the recent progress towards solving the no-dimensional Tverberg problem and new open questions arising in its context. Also, we discuss the colorful variation of this problem and its algorithmic aspects, particularly focusing on the case when each part of a partition contains exactly 2 points. The latter turns out to be related to the following no-dimensional Tverberg-type problem of Huemer et al.: For an even set of points in Euclidean space, find a perfect matching such that the balls with diameters induced by its edges intersect.

math.CO↗
arXiv2024

Tight colorful no-dimensional Tverberg theorem

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 $R$ such that, for any pairwise disjoint $k$-element subsets $Q_1,\dots,Q_n$ of a normed space, there exists a partition of $Q_1\cup\cdots\cup Q_n$ into disjoint transversals $\{P_1,\dots,P_k\}$ for which a ball of radius $R$ intersects the convex hull of each $P_i$ ($1\le i\le k$). Our methods are deterministic and dimension-free, and they are unified by optimizing two functionals: a quadratic \emph{selection} functional whose local maximizers produce a complete system of disjoint transversals, and a convex \emph{intersection} functional that certifies a common point. First, in the Euclidean setting we bound $R$ in terms of the Chebyshev radii (minimal enclosing-ball radii) of the color classes $Q_1,\dots,Q_n$. A key observation is a ``combinatorial'' subadditivity of the squared Chebyshev radius: given sequences $X=(x_1,\dots,x_k)$ and $Y=(y_1,\dots,y_k)$ of points in a Euclidean space, contained in balls of radii $R_X$ and $R_Y$ (not necessarily with the same center), one can reenumerate $Y$ so that the pointwise-sum sequence $Z=(x_1+y_1,\dots,x_k+y_k)$ is contained in a ball of radius $R_Z$ satisfying \[ R_Z^2 \le R_X^2 + R_Y^2 . \] As a corollary, we obtain the best-possible bound \[ R \le \frac{1}{\sqrt{2n}}\sqrt{\frac{k-1}{k}}\, \max_{1\le i\le n} \operatorname{diam}(Q_i). \] Our algorithm returns the desired disjoint transversals in overall time $\mathcal{O}(nk^3)$. Second, we develop a complementary approach based on the inter-color diameter and extend the framework to obtain no-dimensional colorful Tverberg-type results in the hyperbolic setting and in Banach spaces.

math.MG↗