A. Prymak

activity

20102026

most citedUniform and Pointwise Shape Preserving Approximation by Algebraic Polynomials

33 citations · 149 across the 36 of their papers we have counts for

collaborators

38 papers

math.NA2026

Sampling discretization of the uniform norm for hyperbolic-cross trigonometric polynomials

Feng Dai, Andriy Prymak

We study sampling discretization of the uniform norm for trigonometric polynomials with frequencies in a hyperbolic cross of level $N$ on $\T^d$. For every $d, N \geq 2$ and $\varepsil…

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.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.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

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.CA2023

Whitney-type estimates for convex functions

Jaskaran Singh Kaire, Andriy Prymak

We study Whitney-type estimates for approximation of convex functions in the uniform norm on various convex multivariate domains while paying a particular attention to the dependen…

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.NA2022★ 1 cited

On the cardinality of lower sets and universal discretization

F. Dai, A. Prymak, A. Shadrin +2

A set $Q$ in $Z_{+}^{d}$ is a lower set if $(k_{1}, \dots, k_{d}) \in Q$ implies $(l_{1}, \dots, l_{d}) \in Q$ whenever $0 \leq l_{i} \leq k_{i}$ for all $i$ . We derive new and refine known results…

math.CA2022★ 1 cited

Polynomial approximation on $C^{2}$ -domains

Feng Dai, Andriy Prymak

We introduce appropriate computable moduli of smoothness to characterize the rate of best approximation by multivariate polynomials on a connected and compact $C^{2}$ -domain $Ω\subse…

math.NA2022★ 8 cited

Optimal polynomial meshes exist on any multivariate convex domain

Feng Dai, Andriy Prymak

We show that optimal polynomial meshes exist for every convex body in $R^{d}$ , confirming a conjecture by A. Kroo.

math.NA2022★ 4 cited

On Bernstein- and Marcinkiewicz-type inequalities on multivariate $C^{α}$ -domains

Feng Dai, András Kroó, Andriy Prymak

We prove new Bernstein and Markov type inequalities in $L^{p}$ spaces associated with the normal and the tangential derivatives on the boundary of a general compact $C^{α}$ -domain with…

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…

math.MG2021★ 4 cited

Every $3$ -dimensional convex body can be covered by $14$ smaller homothetic copies

A. Prymak

We show that every $3$ -dimensional convex body can be covered by $14$ smaller homothetic copies. The previous result was $16$ copies established by Papadoperakis in 1999, while a c…

math.NA2020★ 2 cited

On directional Whitney inequality

Feng Dai, Andriy Prymak

This paper studies a new Whitney type inequality on a compact domain $Ω \subset R^{d}$ that takes the form $$\inf_{Q\in Π_{r-1}^d({\mathcal{E}})} \|f-Q\|_p \leq C(p,r,Ω) ω_…

math.CA2020★ 11 cited

$L^{p}$ -Bernstein inequalities on $C^{2}$ -domains and applications to discretization

Feng Dai, Andriy Prymak

We prove a new Bernstein type inequality in $L^{p}$ spaces associated with the tangential derivatives on the boundary of a general compact $C^{2}$ -domain. We give two applications: Mar…

math.CA2020★ 2 cited

Geometric computation of Christoffel functions on planar convex domains

A. Prymak

For arbitrary planar convex domain, we compute the behavior of Christoffel function up to a constant factor using comparison with other simple reference domains. The lower bound is…

math.CA2020★ 18 cited

Entropy numbers and Marcinkiewicz-type discretization theorem

F. Dai, A. Prymak, A. Shadrin +2

This paper studies the behavior of the entropy numbers of classes of functions with bounded integral norms from a given finite dimensional linear subspace. Upper bounds of these en…

math.CA2019

Polynomial approximation on $C^{2}$ -domains

Feng Dai, Andriy Prymak

We introduce appropriate computable moduli of smoothness to characterize the rate of best approximation by multivariate polynomials on a connected and compact $C^{2}$ -domain $Ω\subse…

Sampling discretization of the uniform norm for hyperbolic-cross trigonometric polynomials

On Grünbaum's problem for symmetric configurations

Exponential bounds for the spherical Blaschke-Lebesgue problem

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

Small Volume Bodies of Constant Width with Tetrahedral Symmetries

Small volume bodies of constant width

On Hadwiger's covering problem in small dimensions

Whitney-type estimates for convex functions

Convex bodies of constant width with exponential illumination number

On the cardinality of lower sets and universal discretization

Polynomial approximation on C2-domains

Optimal polynomial meshes exist on any multivariate convex domain

On Bernstein- and Marcinkiewicz-type inequalities on multivariate Cα-domains

Upper bounds on chromatic number of En in low dimensions

Every 3-dimensional convex body can be covered by 14 smaller homothetic copies

On directional Whitney inequality

Lp-Bernstein inequalities on C2-domains and applications to discretization

Geometric computation of Christoffel functions on planar convex domains

Entropy numbers and Marcinkiewicz-type discretization theorem

Polynomial approximation on C2-domains

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

Polynomial approximation on $C^{2}$ -domains

On Bernstein- and Marcinkiewicz-type inequalities on multivariate $C^{α}$ -domains

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

Every $3$ -dimensional convex body can be covered by $14$ smaller homothetic copies

$L^{p}$ -Bernstein inequalities on $C^{2}$ -domains and applications to discretization

Polynomial approximation on $C^{2}$ -domains