Alexander Tolmachev

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20242026

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math.MG2026

Reducing the upper bound for the Borsuk number in $R^{4}$ to 8

Alexander Tolmachev, Vsevolod Voronov

The Borsuk number $b (n)$ of $n$ -dimensional Euclidean space $R^{n}$ is the smallest integer such that any set $F \subset R^{n}$ of unit diameter can be partitioned in…

math.MG2025

On lower bounds of the density of planar periodic sets without unit distances

Alexander Tolmachev

Determining the maximal density $m_{1} (R^{2})$ of planar sets without unit distances is a fundamental problem in combinatorial geometry. This paper investigates lower bounds f…

math.MG2024

Optimal partitions of the flat torus into parts of smaller diameter

Dmitry Protasov, Alexander Tolmachev, Vsevolod Voronov

We consider the problem of partitioning a two-dimensional flat torus $T^{2}$ into $m$ sets in order to minimize the maximal diameter of a part. For $m ⩽ 25$ we give numerical…

Reducing the upper bound for the Borsuk number in R4 to 8

On lower bounds of the density of planar periodic sets without unit distances

Optimal partitions of the flat torus into parts of smaller diameter

Reducing the upper bound for the Borsuk number in $R^{4}$ to 8