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Sergii Myroshnychenko

Publications and source records attributed to Sergii Myroshnychenko.

15 recordsLinked to original sources
arXiv2026

On convex bodies with rotationally symmetric planar projections

Let $n\ge 3$ and let $K\subset\mathbb R^n$ be a convex body. For a two-dimensional linear subspace $P\subset\mathbb R^n$, let $K|P$ be the orthogonal projection of $K$ onto $P$. We prove that if for every two-dimensional subspace $P$, the planar convex body $K|P$ has $q$-fold rotational symmetry up to translation, then for $q\ge4$, this forces $K$ to be an Euclidean ball. The case $q=3$ is exceptional: non-spherical examples exist.

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arXiv2026

Area Minimization Among Group-Invariant Planar Convex Bodies of Constant Width

The classical Blaschke--Lebesgue theorem identifies the Reuleaux triangle as the planar convex body of constant width with minimum area. We investigate this extremal problem under prescribed symmetry constraints. Specifically, we classify the minimum-area convex bodies of constant width that are invariant under a finite group $G$ of isometries of $\mathbb{R}^2$ fixing the origin. For the exceptional reflection group $D_1$, the minimizers are precisely the Reuleaux triangles invariant under the prescribed reflection. If $G$ contains the half-turn $\mathcal R_π$, the disk is the unique minimizer. For odd $n\geq 3$, the minimizers are regular Reuleaux $n$-gons, unique up to rotation in the cyclic case $C_n$, and exactly those satisfying the prescribed reflection symmetry in the dihedral case $D_n$.

math.MG↗
arXiv2026

The Spherical Grünbaum Inequality

We prove an analogue of Grünbaum's inequality on the sphere. Let $n \geq 3$ and let $K$ be a convex body on $\mathbb S^{n-1}\subset \mathbb R^n$ with centroid at $θ\in \mathbb S^{n-1}$. Then for any $u\in \mathbb S^{n-1}$ that is orthogonal to $θ$ we have $$σ(K\cap u^+) \ge \left(1-\frac{1}{n}\right)^{n-1} σ(K),$$ where $σ$ denotes the spherical measure. The constant in this inequality is optimal.

math.MG↗
arXiv2026

On hyperbolic and functional analogues of questions of Grünbaum and Loewner

Myroshnychenko, Tatarko, and Yaskin constructed a body $K$ in $\mathbb{R}^n$, $n \geq 5$, with the property that there is exactly one hyperplane $H$ passing through $c(K)$, the centroid of $K$, such that the centroid of $K\cap H$ coincides with $c(K)$. This construction provided answers to questions of Grünbaum and Loewner for $n\geq 5$, which are still open in dimensions $3$ and $4$. We study analogues of these questions in the settings of hyperbolic space $\mathbb H^n$ and $s$-concave functions on $\mathbb R^n$.

math.MG↗
arXiv2026

Uniqueness of Flotation and Buoyancy Surfaces for Convex Polytopes

We prove that a convex polytope $P \subset \mathbb{R}^d$, $d \ge 2$, of uniform density $δ\in (0,1)$ floating in a liquid of density $1$, is uniquely determined by its surface of flotation $P_{[δ]}$ whenever $δ\neq \tfrac{1}{2}$. Analogously, we show that the buoyancy surface $\mathcal{C}_δP$ of a convex polytope $P$ with prescribed density $δ\in (0,1)$ uniquely determines $P$.

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arXiv2023

Entropic exercises around the Kneser-Poulsen conjecture

We develop an information-theoretic approach to study the Kneser--Poulsen conjecture in discrete geometry. This leads us to a broad question regarding whether Rényi entropies of independent sums decrease when one of the summands is contracted by a $1$-Lipschitz map. We answer this question affirmatively in various cases.

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arXiv2022

How far apart can the projection of the centroid of a convex body and the centroid of its projection be?

We show that there is a constant $D \approx 0.2016$ such that for every $n$, every convex body $K\subset \mathbb R^n$, and every hyperplane $H\subset \mathbb R^n$, the distance between the projection of the centroid of $K$ onto $H$ and the centroid of the projection of $K$ onto $H$ is at most $D$ times the width of $K$ in the direction of the segment connecting the two points. The constant $D$ is asymptotically sharp.

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arXiv2021

Functional Response Designs via the Analytic Permutation Test

Vast literature on experimental design extends from Fisher and Snedecor to the modern day. When data lies beyond the assumption of univariate normality, nonparametric methods including rank based statistics and permutation tests are enlisted. The permutation test is a versatile exact nonparametric significance test that requires drastically fewer assumptions than similar parametric tests. The main downfall of the permutation test is high computational cost making this approach laborious for complex data and sophisticated experimental designs and completely infeasible in any application requiring speedy results such as high throughput streaming data. We rectify this problem through application of concentration inequalities and thus propose a computation free permutation test -- i.e. a permutation-less permutation test. This general framework is applied to multivariate, matrix-valued, and functional data. We improve these concentration bounds via a novel incomplete beta transform. We extend our theory from 2-sample to $k$-sample testing through the use of weakly dependent Rademacher chaoses and modified decoupling inequalities. We test this methodology on classic functional data sets including the Berkeley growth curves and the phoneme dataset. We further consider analysis of spoken vowel sound under two experimental designs: the Latin square and the randomized block design.

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arXiv2020

Unique determination of ellipsoids by their dual volumes and the moment problem

Gusakova and Zaporozhets conjectured that ellipsoids in $\mathbb R^n$ are uniquely determined (up to an isometry) by their Steiner polynomials. Petrov and Tarasov confirmed this conjecture in $\mathbb R^3$. In this paper we solve the dual problem. We show that any ellipsoid in $\mathbb{R}^n$ centered at the origin is uniquely determined (up to an isometry) by its dual Steiner polynomial. To prove this result we reduce it to a problem of moments. As a by-product we give an alternative proof of the result of Petrov and Tarasov.

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arXiv2018

On recognizing shapes of polytopes from their shadows

Let $P$ and $Q$ be two convex polytopes both contained in the interior of an Euclidean ball $r\textbf{B}^{d}$. We prove that $P=Q$ provided that their sight cones from any point on the sphere $rS^{d-1}$ are congruent. We also prove an analogous result for spherical projections.

math.MG↗
arXiv2017

Grünbaum's inequality for sections

We show \begin{align*} \frac{ \int_{E \cap θ^+} f(x) dx }{ \int_E f(x) dx } \geq \left(\frac{k γ+1}{(n+1) γ+1}\right)^{\frac{k γ+1}γ} \end{align*} for all $k$-dimensional subspaces $E\subset\mathbb{R}^n$, $θ\in E\cap S^{n-1}$, and all $γ$-concave functions $f:\mathbb{R}^n\rightarrow [0,\infty)$ with $γ>0$, $0< \int_{\mathbb{R}^n} f(x)\, dx <\infty$, and $\int_{\mathbb{R}^n} x f(x)\, dx$ at the origin $o\in\mathbb{R}^n$. Here, $θ^+ := \lbrace x\, : \, \langle x,θ\rangle \geq 0 \rbrace$. As a consequence of this result, we get the following generalization of Grünbaum's inequality: \begin{align*} \frac{ \mbox{vol}_k(K\cap E\capθ^+) }{ \mbox{vol}_k(K\cap E) } \geq \left( \frac{k}{n+1} \right)^k \end{align*} for all convex bodies $K\subset\mathbb{R}^n$ with centroid at the origin, $k$-dimensional subspaces $E\subset\mathbb{R}^n$, and $θ\in E\cap S^{n-1}$. The lower bounds in both of our inequalities are the best possible, and we discuss the equality conditions.

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arXiv2017

On polytopes with congruent projections or sections

Let $2\le k\le d-1$ and let $P$ and $Q$ be two convex polytopes in ${\mathbb E^d}$. Assume that their projections, $P|H$, $Q|H$, onto every $k$-dimensional subspace $H$, are congruent. In this paper we show that $P$ and $Q$ or $P$ and $-Q$ are translates of each other. We also prove an analogous result for sections by showing that $P=Q$ or $P=-Q$, provided the polytopes contain the origin in their interior and their sections, $P \cap H$, $Q \cap H$, by every $k$-dimensional subspace $H$, are congruent.

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arXiv2016

Star bodies with completely symmetric sections

We say that a star body $K$ is completely symmetric if it has centroid at the origin and its symmetry group $G$ forces any ellipsoid whose symmetry group contains $G$, to be a ball. In this short note, we prove that if all central sections of a star body $L$ are completely symmetric, then $L$ has to be a ball. A special case of our result states that if all sections of $L$ are origin symmetric and 1-symmetric, then $L$ has to be a Euclidean ball. This answers a question from \cite{R2}. Our result is a consequence of a general theorem that we establish, stating that if the restrictions in almost all equators of a real function $f$ defined on the sphere, are isotropic functions, then $f$ is constant a.e. In the last section of this note, applications, improvements and related open problems are discussed and two additional open questions from \cite{R} and \cite{R2} are answered.}

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arXiv2016

On a functional equation related to a pair of hedgehogs with congruent projections

Hedgehogs are geometrical objects that describe the Minkowski differences of arbitrary convex bodies in the Euclidean space $\mathbb{E}^n$. We prove that two hedgehogs in $\mathbb{E}^n, n \geq 3$, coincide up to a translation and a reflection in the origin, provided that their projections onto any two-dimensional plane are directly congruent and have no direct rigid motion symmetries. Our result is a consequence of a more general analytic statement about the solutions of a functional equation in which the support functions of hedgehogs are replaced with two arbitrary twice continuously differentiable functions on the unit sphere.

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