Andrii Arman

activity

20212026

most citedSmall volume bodies of constant width

1 citations · 2 across the 11 of their papers we have counts for

collaborators

12 papers

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.CO2026

A construction of spherical $5$ -designs with $O (d^{2})$ points

Andrii Arman, Andriy Bondarenko, Andriy Prymak +1

For every $d \geq 1$ we give an explicit equal-weight spherical $5$ -design in $S^{d - 1} \subset R^{d}$ with at most $72 d^{2}$ points. Our approach utilizes recent construc…

math.MG2025

On asymptotic Lebesgue's universal covering problem

Andrii Arman, Andriy Bondarenko, Andriy Prymak +1

A 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 th…

math.MG2025

Hadwiger's conjecture for cap bodies

Andrii Arman, Jaskaran Singh Kaire, Andriy Prymak

Hadwiger's covering conjecture states that every $n$ -dimensional convex body can be covered by at most $2^{n}$ of its smaller positive homothetic translates, with $2^{n}$ copies requir…

math.MG2025

Illumination number of 3-dimensional cap bodies

Andrii Arman, Jaskaran Singh Kaire, Andriy Prymak

The illumination conjecture asserts that any convex body in $n$ -dimensional Euclidean space can be illuminated by at most $2^{n}$ external light sources or parallel beams of light. D…

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

Minimal dispersion on the cube and the torus

Andrii Arman, Alexander E. Litvak

We improve some upper bounds for minimal dispersion on the cube and torus. /Our new ingredient is an improvement of a probabilistic lemma used to obtain upper bounds for dispersion…

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★ 1 cited

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…

math.MG2023

Convex bodies of constant width with exponential illumination number

Andrii Arman, Andriy Bondarenko, Andriy Prymak

We show that there exist convex bodies of constant width in $E^{n}$ with illumination number at least $(cos (π /14) + o (1))^{- n}$ , answering a question by G. Kalai. Furthermore…

math.CO2021★ 1 cited

Upper bounds on chromatic number of $E^{n}$ in low dimensions

Andrii Arman, Andriy V. Bondarenko, Andriy Prymak +1

Let $χ (E^{n})$ denote the chromatic number of the Euclidean space $E^{n}$ , i.e., the smallest number of colors that can be used to color $E^{n}$ so that no two…

On Grünbaum's problem for symmetric configurations

A construction of spherical 5-designs with O(d2) points

On asymptotic Lebesgue's universal covering problem

Hadwiger's conjecture for cap bodies

Illumination number of 3-dimensional cap bodies

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

Minimal dispersion on the cube and the torus

Small Volume Bodies of Constant Width with Tetrahedral Symmetries

Small volume bodies of constant width

On Hadwiger's covering problem in small dimensions

Convex bodies of constant width with exponential illumination number

Upper bounds on chromatic number of En in low dimensions

A construction of spherical $5$ -designs with $O (d^{2})$ points

Upper bounds on chromatic number of $E^{n}$ in low dimensions