arXiv subjects

Alexandr Polyanskii

Publications and source records attributed to Alexandr Polyanskii.

23 records · Page 2Linked to original sources

arXiv2017

Helly-type theorem for eigenvectors

We prove that if any $\lfloor3d/2 \rfloor$ or fewer elements of a finite family of linear operators $\mathbb K^d\to \mathbb K^d$ ($\mathbb K$ is an arbitrary field) have a common eigenvector then all operators in the family have a common eigenvector. Moreover, $\lfloor 3d/2\rfloor$ cannot be replaced by a smaller number. Also, we study the following problem, achieving partial results: prove that if any $l=O(d)$ or fewer elements of a finite family of linear operators $\mathbb K^d\to \mathbb K^d$ have a common non-trivial invariant subspace then all operators in the family have a common non-trivial invariant subspace.

math.MG↗

arXiv2017

On reduced polytopes

A convex body $R$ in $\mathbb R^d$ is called reduced if the minimal width $Δ(R')$ of each convex body $R'\subset R$ different from $R$ is strictly smaller than the minimal width $Δ(R)$ of $R$. In this article we construct a reduced polytope in $\mathbb R^3$, i.e. we answer the following question posed by Lassak: do there exist reduced polytopes in $\mathbb R^d$, $d\geqslant3$? Also, we prove some properties of reduced polytopes in $\mathbb R^3$.

math.MG↗

arXiv2017

Hunting for reduced polytopes

We show that there exist reduced polytopes in three-dimensional Euclidean space. This partially answers the question posed by Lassak on the existence of reduced polytopes in $d$-dimensional Euclidean space for $d\geq 3$. Moreover, we prove a novel necessary condition on reduced polytopes in three-dimensional Euclidean space.

math.MG↗

arXiv2016

Proof of Schur's conjecture in $\mathbb R^d$

In this paper we prove Schur's conjecture in $\mathbb R^d$, which states that any diameter graph $G$ in the Euclidean space $\mathbb R^d$ on $n$ vertices may have at most $n$ cliques of size $d$. We obtain an analogous statement for diameter graphs with unit edge length on a sphere $S^d_r$ of radius $r>1/\sqrt 2$. The proof rests on the following statement, conjectured by F. Morić and J. Pach: given two unit regular simplices $Δ_1,Δ_2$ on $d$ vertices in $\mathbb R^d$, either they share $d-2$ vertices, or there are vertices $v_1\in Δ_1,v_2\in Δ_2$ such that $\|v_1-v_2\|>1$. The same holds for unit simplices on a $d$-dimensional sphere of radius greater than $1/\sqrt 2$.

math.MG↗

arXiv2015

On the irrationality measure of certain numbers

The paper presents upper estimates for the irrationality measure and the non-quadraticity measure for the numbers $α_k=\sqrt{2k+1}\ln\frac{\sqrt{2k+1}-1}{\sqrt{2k+1}+1}, \ k\in\mathbb N.$

math.NT↗