A. Prymak

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20242026

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math.MG2026

On GrÃ¼nbaum'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.MG2026

Exponential bounds for the spherical Blaschke-Lebesgue problem

Abigail Hall, Andriy Prymak, Chanatip Sujsuntinukul

The Blaschke-Lebesgue theorem states that the Reuleaux triangle has the smallest area among planar convex bodies of a fixed constant width. We study how small bodies of constant wi…

math.MG2026

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.MG2025

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.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

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…