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Alexey Balitskiy

Publications and source records attributed to Alexey Balitskiy.

16 recordsLinked to original sources
arXiv2026

Triangle covering problems and the Viterbo inequality in the plane

We review a certain problem on covering triangles in the plane. Equivalently, it can be viewed as a family of 'isobilliard' inequalities in convex shapes, and as a special case of Viterbo's conjecture in symplectic geometry. We give an elementary overview of these topics and, using the optics of the covering problem, we establish several new special cases of Viterbo's conjecture, provide a simple explanation of the counterexample of Haim-Kislev and Ostrover, and state a few open questions. The main novel result is a proof of Viterbo's conjecture for lagrangian products $K \times Q$, where $Q \subset \mathbb{R}^2$ is any quadrilateral and $K \subset \mathbb{R}^2$ is any convex shape.

math.MG↗
arXiv2025

Geometric Bounds for Persistence

In this paper, we offer a new perspective on persistent homology by integrating key concepts from metric geometry. For a given compact subset $\mathcal{X}$ of a Banach space $\mathbf{Y}$, we analyze the topological features arising in the family $\mathcal{N}_\bullet(\mathcal{X} \subset \mathbf{Y})$ of nested neighborhoods of $\mathcal{X}$ in $\mathbf{Y}$ and provide several geometric bounds on their persistence (lifespans). We begin by examining the lifespans of these homology classes in terms of their filling radii in $\mathbf{Y}$, establishing connections between these lifespans and fundamental invariants in metric geometry, such as the Urysohn width. We then derive bounds on these lifespans by considering the $\ell^\infty$-principal components of $\mathcal{X}$, also known as Kolmogorov widths. Additionally, we introduce and investigate the concept of extinction time of a metric space $\mathcal{X}$: the critical threshold beyond which no homological features persist in any degree. We propose methods for estimating the Čech and Vietoris-Rips extinction times of $\mathcal{X}$ by relating $\mathcal{X}$ to its convex hull and to its tight span, respectively.

math.AT↗
arXiv2023

Systolic almost-rigidity modulo 2

No power law systolic freedom is possible for the product of mod $2$ systoles of dimension $1$ and codimension $1$. This means that any closed $n$-dimensional Riemannian manifold $M$ of bounded local geometry obeys the following systolic inequality: the product of its mod $2$ systoles of dimensions $1$ and $n-1$ is bounded from above by $c(n,\varepsilon) \mbox{Vol}(M)^{1+\varepsilon}$, if finite (if $H_1(M; \mathbb{Z}/2)$ is non-trivial).

math.DG↗
arXiv2022

Systolic inequalities for the number of vertices

Inspired by the classical Riemannian systolic inequality of Gromov we present a combinatorial analogue providing a lower bound on the number of vertices of a simplicial complex in terms of its edge-path systole. Similarly to the Riemannian case, where the inequality holds under a topological assumption of "essentiality", our proofs rely on a combinatorial analogue of that assumption. Under a stronger assumption, expressed in terms of cohomology cup-length, we improve our results quantitatively. We also illustrate our methods in the continuous setting, generalizing and improving quantitatively the Minkowski principle of Balacheff and Karam; a corollary of this result is the extension of the Guth--Nakamura cup-length systolic bound from manifolds to complexes.

math.MG↗
arXiv2021

Macroscopic scalar curvature and codimension 2 width

We show that a complete $3$-dimensional Riemannian manifold $M$ with finitely generated first homology has macroscopic dimension $1$ if it satisfies the following "macroscopic curvature" assumptions: every ball of radius $10$ in $M$ has volume at most $4$, and every loop in every ball of radius $1$ in $M$ is null-homologous in the concentric ball of radius $2$.

math.DG↗
arXiv2021

Waist of maps measured via Urysohn width

We discuss various questions of the following kind: for a continuous map $X \to Y$ from a compact metric space to a simplicial complex, can one guarantee the existence of a fiber large in the sense of Urysohn width? The $d$-width measures how well a space can be approximated by a $d$-dimensional complex. The results of this paper include the following. 1) Any piecewise linear map $f: [0,1]^{m+2} \to Y^m$ from the unit euclidean $(m+2)$-cube to an $m$-polyhedron must have a fiber of $1$-width at least $\frac{1}{2βm +m^2 + m + 1}$, where $β= \sup_y \text{ rk } H_1(f^{-1}(y))$ measures the topological complexity of the map. 2) There exists a piecewise smooth map $X^{3m+1} \to \mathbb{R}^m$, with $X$ a riemannian $(3m+1)$-manifold of large $3m$-width, and with all fibers being topological $(2m+1)$-balls of arbitrarily small $(m+1)$-width.

math.MG↗
arXiv2021

Local-to-global Urysohn width estimates

The notion of the Urysohn $d$-width measures to what extent a metric space can be approximated by a $d$-dimensional simplicial complex. We investigate how local Urysohn width bounds on a riemannian manifold affect its global width. We bound the $1$-width of a Riemannian manifold in terms of its first homology and the supremal width of its unit balls. Answering a question of Larry Guth, we give examples of $n$-manifolds of considerable $(n-1)$-width in which all unit balls have arbitrarily small $1$-width. We also give examples of topologically simple manifolds that are locally nearly low-dimensional.

math.MG↗
arXiv2020

A multi-plank generalization of the Bang and Kadets inequalities

If a convex body $K \subset \mathbb{R}^n$ is covered by the union of convex bodies $C_1, \ldots, C_N$, multiple subadditivity questions can be asked. Two classical results regard the subadditivity of the width (the smallest distance between two parallel hyperplanes that sandwich $K$) and the inradius (the largest radius of a ball contained in $K$): the sum of the widths of the $C_i$ is at least the width of $K$ (this is the plank theorem of Thoger Bang), and the sum of the inradii of the $C_i$ is at least the inradius of $K$ (this is due to Vladimir Kadets). We adapt the existing proofs of these results to prove a theorem on coverings by certain generalized non-convex "multi-planks". One corollary of this approach is a family of inequalities interpolating between Bang's theorem and Kadets's theorem. Other corollaries include results reminiscent of the Davenport--Alexander problem, such as the following: if an $m$-slice pizza cutter (that is, the union of $m$ equiangular rays in the plane with the same endpoint) in applied $N$ times to the unit disk, then there will be a piece of the partition of inradius at least $\frac{\sin π/m}{N + \sin π/m}$.

math.MG↗
arXiv2020

Flip cycles in plabic graphs

Planar bicolored (plabic) graphs are combinatorial objects introduced by Postnikov to give parameterizations of the positroid cells of the totally nonnegative Grassmannian $\text{Gr}^{\geq 0}(n,k)$. Any two plabic graphs for the same positroid cell can be related by a sequence of certain moves. The flip graph has plabic graphs as vertices and has edges connecting the plabic graphs which are related by a single move. A recent result of Galashin shows that plabic graphs can be seen as cross-sections of zonotopal tilings for the cyclic zonotope $Z(n,3)$. Taking this perspective, we show that the fundamental group of the flip graph is generated by cycles of length 4, 5, and 10, and use this result to prove a related conjecture of Dylan Thurston about triple crossing diagrams. We also apply our result to make progress on an instance of the generalized Baues problem.

math.CO↗
arXiv2018

Equality cases in Viterbo's conjecture and isoperimetric billiard inequalities

In this note we apply the billiard technique to deduce some results on Viterbo's conjectured inequality between volume of a convex body and its symplectic capacity. We show that the product of a permutohedron and a simplex (properly related to each other) delivers equality in Viterbo's conjecture. Using this result as well as previously known equality cases, we prove some special cases of Viterbo's conjecture and interpret them as isoperimetric-like inequalities for billiard trajectories.

math.MG↗
arXiv2017

On the Circle Covering Theorem by A. W. Goodman and R. E. Goodman

In 1945, A. W. Goodman and R. E. Goodman proved the following conjecture by P. Erdős: Given a family of (round) disks of radii $r_1$, $\ldots$, $r_n$ in the plane it is always possible to cover them by a disk of radius $R = \sum r_i$, provided they cannot be separated into two subfamilies by a straight line disjoint from the disks. In this note we show that essentially the same idea may work for different analogues and generalizations of their result. In particular, we prove the following: Given a family of positive homothetic copies of a fixed convex body $K \subset \mathbb{R}^d$ with homothety coefficients $τ_1, \ldots, τ_n > 0$ it is always possible to cover them by a translate of $\frac{d+1}{2}\left(\sum τ_i\right)K$, provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets.

math.MG↗
arXiv2017

Optimality of codes with respect to error probability in Gaussian noise

We consider geometrical optimization problems related to optimizing the error probability in the presence of a Gaussian noise. One famous questions in the field is the "weak simplex conjecture". We discuss possible approaches to it, and state related conjectures about the Gaussian measure, in particular, the conjecture about minimizing of the Gaussian measure of a simplex. We also consider antipodal codes, apply the Šidák inequality and establish some theoretical and some numerical results about their optimality.

math.MG↗
arXiv2016

Dependence of the heavily covered point on parameters

We examine Gromov's method of selecting a point "heavily covered" by simplices formed by a given finite point sets, in order to understand the dependence of the heavily covered point on parameters. We have no continuous dependence, but manage to utilize the "homological continuous dependence" of the heavily covered point. This allows us to infer some corollaries in a usual way. We also give an elementary argument to prove the simplest of these corollaries.

math.MG↗
arXiv2015

Billiards in convex bodies with acute angles

In this paper we investigate the existence of closed billiard trajectories in not necessarily smooth convex bodies. In particular, we show that if a body $K\subset \mathbb{R}^d$ has the property that the tangent cone of every non-smooth point $q\in \partial K$ is acute (in a certain sense) then there is a closed billiard trajectory in $K$.

math.MG↗
arXiv2013

Another ham sandwich in the plane

We show that every two nice measures in the plane can be partitioned into equal halves by translation of an angle from arbitrary k-fan when k is odd and in some cases when k is even. We also give some counterexamples for certain fans and measures.

math.MG↗