A. Bondarenko

activity

20232026

collaborators

Showing math.MGShow all

6 papers · 1 filter

math.MG2026

On Grünbaum's problem for symmetric configurations

Andrii Arman, Andriy Bondarenko, Andriy Prymak +1

Let $g_{n}$ be the largest number of Euclidean balls of diameter $1$ which may be needed to cover a set of diameter $1$ in $R^{n}$ . We study this problem for finite sets invar…

math.MG2025

On asymptotic Lebesgue's universal covering problem

Andrii Arman, Andriy Bondarenko, Andriy Prymak +1

Universal cover in $E^{n}$ is a measurable set that contains a congruent copy of any set of diameter 1. Lebesgue's universal covering problem, posed in 1914, asks for the…

math.MG2024

On a Gallai-type problem and illumination of spiky balls and cap bodies

Andrii Arman, Andriy Bondarenko, Andriy Prymak +1

We show that any finite family of pairwise intersecting balls in $E^{n}$ can be pierced by $(3/2 + o (1))^{n}$ points improving the previously known estimate of $(2+o(1))^…

math.MG2024

Small Volume Bodies of Constant Width with Tetrahedral Symmetries

Andrii Arman, Andriy Bondarenko, Andriy Prymak +1

For every $n \geq 2$ , we construct a body $U_{n}$ of constant width $2$ in $E^{n}$ with small volume and symmetries of a regular $n$ -simplex. $U_{2}$ is the Reuleaux triangle. To…

math.MG2024

Small volume bodies of constant width

Andrii Arman, Andriy Bondarenko, Fedor Nazarov +2

For every large enough $n$ , we explicitly construct a body of constant width $2$ that has volume less than $0. 9^{n} Vol (B^{n}$ ), where $B^{n}$ is the unit ba…

math.MG2024

On Hadwiger's covering problem in small dimensions

Andrii Arman, Andriy Bondarenko, Andriy Prymak

Let $H_{n}$ be the minimal number such that any $n$ -dimensional convex body can be covered by $H_{n}$ translates of interior of that body. Similarly $H_{n}^{s}$ is the corresponding quanti…