arXiv subjects

Andriy Prymak

Publications and source records attributed to Andriy Prymak.

At least 19 recordsLinked to original sources

arXiv2026

On Gr\"unbaum's problem for symmetric configurations

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 $\mathbb{R}^n$. We study this problem for finite sets invariant under all coordinate permutations. We prove that the exponential growth rate in this symmetric problem can be characterized exactly as a finite-alphabet squared-error rate-distortion supremum $\alpha_0$. Specialized to the two-point case, i.e., for subsets of Boolean cubes, this gives the explicit lower bound \[g_n\ge (1.160235457\ldots-o(1))^n,\] improving the previous best bound $(2/\sqrt3-o(1))^n$. Using Fix's Gaussian characterization of the rate-distortion problem, we give a numerical three-point construction with exponent base greater than $1.160497831$. Finally, we show that $\alpha_0$ is not attained by any finitely supported distribution.

math.MG↗

arXiv2026

Exponential bounds for the spherical Blaschke-Lebesgue problem

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 width can be on the unit sphere $\mathbb S^n$ when $n$ is large. For a spherical convex body $K\subset \mathbb S^n$ of constant width $w\in(0,\pi)$, its relative effective radius is \[ \left(\frac{\mu_n(K)}{\mu_n(\mathbb B^n(w/2))}\right)^{1/n}, \] where $\mu_n$ is the spherical $n$-measure and $\mathbb B^n(w/2)$ is a geodesic ball of radius $w/2$. Let $\sigma_n(w)$ be the infimum of the relative effective radius over all spherical bodies of constant width $w$. Define $\underline{\sigma}(w)=\liminf_{n\to\infty}\sigma_n(w)$ and $\overline{\sigma}(w)=\limsup_{n\to\infty}\sigma_n(w)$. For each fixed $w\in(0,\pi)$, we prove non-trivial bounds \[ 0<\sigma_{\ell}(w)\le \underline{\sigma}(w)\le \overline{\sigma}(w)\le \sigma_u(w)<1, \] where $\sigma_\ell(w)$ and $\sigma_u(w)$ are defined in terms of $w$ either explicitly or through a root of a quartic equation. The upper bounds are obtained by constructing small spherical bodies of constant width: for $w\le\pi/2$ by a spherical version of the recent Arman-Bondarenko-Nazarov-Prymak-Radchenko Euclidean example, and for $w>\pi/2$ by spherical duality. The lower bounds combine a spherical adaptation of Schramm's illumination argument with a Gaussian autocorrelation method for the associated convex cones.

math.MG↗

arXiv2026

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

For every $d\geq1$ we give an explicit equal-weight spherical $5$-design in $\mathbb{S}^{d-1}\subset\mathbb{R}^d$ with at most $72d^2$ points. Our approach utilizes recent construction of complex projective $2$-designs based on Sidon sets.

math.CO↗

arXiv2025

On asymptotic Lebesgue's universal covering problem

Universal cover in $\mathbb{E}^{n}$ 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 convex set of smallest area that serves as a universal cover in the plane ($n=2$). A simple universal cover in $\mathbb{E}^n$ is provided by the classical theorem of Jung, which states that any set of diameter 1 in an $n$-dimensional Euclidean space is contained in a ball $J_n$ of radius $\sqrt{\tfrac{n}{2n+2}}$; in other words, $J_n$ is a universal cover in $\mathbb{E}^n$. We show that in high dimensions, Jung's ball $J_n$ is asymptotically optimal with respect to the volume, namely, for any universal cover $U \subset \mathbb{E}^n$, $$ {\rm Vol}(U) \ge (1-o(1))^n{\rm Vol}(J_n). $$

math.MG↗

arXiv2025

Hadwiger's conjecture for cap bodies

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 required only for affine images of the $n$-cube. In this note, we confirm Hadwiger's conjecture for the class of cap bodies in all dimensions, bridging recently established cases of $n=3$ and large $n$. The proof uses probabilistic techniques and, for $4\le n \le 15$, computer-assisted linear programming.

math.MG↗

arXiv2025

Illumination number of 3-dimensional cap bodies

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. Despite recent progress on the illumination conjecture, it remains open in general, as well as for specific classes of bodies. Bezdek, Ivanov, and Strachan showed that the conjecture holds for symmetric cap bodies in sufficiently high dimensions. Further, Ivanov and Strachan calculated the illumination number for the class of 3-dimensional centrally symmetric cap bodies to be 6. In this paper, we show that even the broader class of all 3-dimensional cap bodies has the same illumination number 6, in particular, the illumination conjecture holds for this class. The illuminating directions can be taken to be vertices of a regular tetrahedron, together with two special directions depending on the body. The proof is based on probabilistic arguments and integer linear programming.

math.MG↗

arXiv2024

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

We show that any finite family of pairwise intersecting balls in $\mathbb{E}^n$ can be pierced by $(\sqrt{3/2}+o(1))^n$ points improving the previously known estimate of $(2+o(1))^n$. As a corollary, this implies that any $2$-illuminable spiky ball in $\mathbb{E}^n$ can be illuminated by $(\sqrt{3/2}+o(1))^n$ directions. For the illumination number of convex spiky balls, i.e., cap bodies, we show an upper bound in terms of the sizes of certain related spherical codes and coverings. For large dimensions, this results in an upper bound of $1.19851^n$, which can be compared with the previous $(\sqrt{2}+o(1))^n$ established only for the centrally symmetric cap bodies. We also prove the lower bounds of $(\tfrac{2}{\sqrt{3}}-o(1))^n$ for the three problems above.

math.MG↗

arXiv2024

Small Volume Bodies of Constant Width with Tetrahedral Symmetries

For every $n\ge 2$, we construct a body $U_n$ of constant width $2$ in $\mathbb{E}^n$ with small volume and symmetries of a regular $n$-simplex. $U_2$ is the Reuleaux triangle. To the best of our knowledge, $U_3$ was not previously constructed, and its volume is smaller than the volume of other three-dimensional bodies of constant width with tetrahedral symmetries. While the volume of $U_3$ is slightly larger than the volume of Meissner's bodies of width $2$, it exceeds the latter by less than $0.137\%$. For all large $n$, the volume of $U_n$ is smaller than the volume of the ball of radius $0.891$.

math.MG↗

arXiv2024

Small volume bodies of constant width

For every large enough $n$, we explicitly construct a body of constant width $2$ that has volume less than $0.9^n \text{Vol}(\mathbb{B}^{n}$), where $\mathbb{B}^{n}$ is the unit ball in $\mathbb{R}^{n}$. This answers a question of O.~Schramm.

math.MG↗

arXiv2024

On Hadwiger's covering problem in small dimensions

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 quantity for symmetric bodies. It is possible to define $H_n$ and $H_n^s$ in terms of illumination of the boundary of the body using external light sources, and the famous Hadwiger's covering conjecture (illumination conjecture) states that $H_n=H_{n}^s=2^n$. In this note we obtain new upper bounds on $H_n$ and $H_{n}^s$ for small dimensions $n$. Our main idea is to cover the body by translates of John's ellipsoid (the inscribed ellipsoid of the largest volume). Using specific lattice coverings, estimates of quermassintegrals for convex bodies in John's position, and calculations of mean widths of regular simplexes, we prove the following new upper bounds on $H_n$ and $H_n^s$: $H_5\le 933$, $H_6\le 6137$, $H_7\le 41377$, $H_8\le 284096$, $H_4^s\le 72$, $H_5^s\le 305$, and $H_6^s\le 1292$. For larger $n$, we describe how the general asymptotic bounds $H_n\le \binom{2n}{n}n(\ln n+\ln\ln n+5)$ and $H_n^s\le 2^n n(\ln n+\ln\ln n+5)$ due to Rogers and Shephard can be improved for specific values of $n$.

math.MG↗

arXiv2023

Whitney-type estimates for convex functions

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 dependence of the involved constants on the dimension and the geometry of the domain.

math.CA↗

arXiv2023

Convex bodies of constant width with exponential illumination number

We show that there exist convex bodies of constant width in $\mathbb{E}^n$ with illumination number at least $(\cos(\pi/14)+o(1))^{-n}$, answering a question by G. Kalai. Furthermore, we prove the existence of finite sets of diameter $1$ in $\mathbb{E}^n$ which cannot be covered by $(2/\sqrt{3}+o(1))^{n}$ balls of diameter $1$, improving a result by J. Bourgain and J. Lindenstrauss.

math.MG↗

arXiv2022

Polynomial approximation on $C^2$-domains

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 $\Omega\subset \mathbb{R}^d$. This new modulus of smoothness is defined via finite differences along the directions of coordinate axes, and along a number of tangential directions from the boundary. With this modulus, we prove both the direct Jackson inequality and the corresponding inverse for the best polynomial approximation in $L_p(\Omega)$. The Jackson inequality is established for the full range of $0<p\leq \infty$, while its proof relies on a recently established Whitney type estimates with constants depending only on certain parameters; and on a highly localized polynomial partitions of unity on a $C^2$-domain which is of independent interest. The inverse inequality is established for $1\leq p\leq \infty$, and its proof relies on a recently proved Bernstein type inequality associated with the tangential derivatives on the boundary of $\Omega$. Such an inequality also allows us to establish the inverse theorem for Ivanov's average moduli of smoothness on general compact $C^2$-domains.

math.CA↗

arXiv2022

Optimal polynomial meshes exist on any multivariate convex domain

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

math.NA↗

arXiv2022

On Bernstein- and Marcinkiewicz-type inequalities on multivariate $C^\alpha$-domains

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 $C^\alpha$-domain with $1\leq \alpha\leq 2$. These estimates are also applied to establish Marcinkiewicz type inequalities for discretization of $L^p$ norms of algebraic polynomials on $C^\alpha$-domains with asymptotically optimal number of function samples used.

math.NA↗

arXiv2021

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

Let $\chi(\mathbb{E}^n)$ denote the chromatic number of the Euclidean space $\mathbb{E}^n$, i.e., the smallest number of colors that can be used to color $\mathbb{E}^n$ so that no two points unit distance apart are of the same color. We present explicit constructions of colorings of $\mathbb{E}^n$ based on sublattice coloring schemes that establish the following new bounds: $\chi(\mathbb{E}^5)\le 140$, $\chi(\mathbb{E}^n)\le 7^{n/2}$ for $n\in\{6,8,24\}$, $\chi(\mathbb{E}^7)\le 1372$, $\chi(\mathbb{E}^{9})\leq 17253$, and $\chi(\mathbb{E}^n)\le 3^n$ for all $n\le 38$ and $n=48,49$.

math.CO↗

arXiv2021

On directional Whitney inequality

This paper studies a new Whitney type inequality on a compact domain $Ω\subset {\mathbb{R}}^d$ that takes the form $$\inf_{Q\in Π_{r-1}^d({\mathcal{E}})} \|f-Q\|_p \leq C(p,r,Ω) ω_{\mathcal{E}}^r(f,{\rm diam}(Ω))_p,\ \ r\in {\mathbb{N}},\ \ 0 0$. It is proved that ${\mathcal{N}}_d(Ω)=d$ for every connected $C^2$-domain $Ω\subset {\mathbb{R}}^d$, for $d=2$ and every planar convex body $Ω\subset {\mathbb{R}}^2$, and for $d\ge 3$ and every almost smooth convex body $Ω\subset {\mathbb{R}}^d$. [See the pre-print for the complete abstract - not included here due to arXiv limitations.]

math.NA↗

arXiv2020

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

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: Marcinkiewicz type inequality for discretization of $L^p$ norm and positive cubature formula. Both results are optimal in the sense that the number of function samples used has the order of the dimension of the corresponding space of algebraic polynomials.

math.CA↗