arXiv subjects

Egor Bakaev

Publications and source records attributed to Egor Bakaev.

7 recordsLinked to original sources

arXiv2026

Optimal partial plank coverings

A plank of width $w$ in a Euclidean space is the set of points lying between two parallel hyperplanes at distance $w$ from each other. Bang's theorem says that if a family of planks covers a convex body $K$, then their total width is at least the width of $K$, that is, the width of the thinnest plank containing $K$. We study a quantitative variant of this problem in the case where the total width of the planks is fixed. How should the planks be placed so as to cover as much of the volume of the body as possible? For the central case where $K$ is a Euclidean ball, K\'aroly Bezdek asked whether the optimal arrangement consists of a single plank centered at the origin. We give an affirmative answer to this question. We also show that for every planar convex body an optimal partial covering is attained by a single plank.

math.MG↗

arXiv2026

A simplex-based measure of symmetry

For compact convex sets $L,K \subset \mathbb{R}^n$, denote by $\lambda_K(L)$ the smallest size of a homothet of $K$ that contains $L$. We define a measure of symmetry based on the $n$-simplex $\Delta = \Delta^n \subset \mathbb{R}^n$ as the ratio \[ \rho_\Delta(L):=\frac{\lambda_{-\Delta}(L)}{\lambda_{\Delta}(L)}. \] We study this measure and deduce the following results: (1) The classical Minkowski measure of symmetry $m^*(L)$ can be defined as an affine-invariant version of $\rho_\Delta(L)$. (2) We improve the stability analysis for the Minkowski measure of symmetry; if $m^*(L)\ge n-\varepsilon$ then $L$ is $\tfrac{1}{1-\varepsilon}$-close to $\Delta$ in the Banach--Mazur distance. (3) We obtain a novel characterization of simplices as the only convex bodies $K$ for which the function $L \mapsto \lambda_K(L)$ is additive (a property we term ``outer additivity''). (4) Motivated by the expressivity of ReLU neural networks, we study the depth complexity of polytopes in $\mathbb{R}^n$ under the two operations: Minkowski sum and convex hull of a union. We prove the sharp bound $\rho_\Delta(P) \leq 2^d -1$ for every polytope $P$ of depth complexity $d$. In other words, simplices cannot be approximated by low-depth polytopes.

math.MG↗

arXiv2026

A note on the affine plank conjecture

In 1951, Bang posed the affine plank conjecture, which remains open: If a convex body in $\mathbb{R}^d$ is covered by planks, then the total relative width of the planks is at least one. We prove a lower bound of $2/(1+\sqrt{d})$ for this total relative width. The best previously known lower bound was $2/(1+d)$.

math.MG↗

arXiv2025

Approximation Depth of Convex Polytopes

We study approximations of polytopes in the standard model for computing polytopes using Minkowski sums and (convex hulls of) unions. Specifically, we study the ability to approximate a target polytope by polytopes of a given depth. Our main results imply that simplices can only be ``trivially approximated''. On the way, we obtain a characterization of simplices as the only ``outer additive'' convex bodies.

math.MG↗

arXiv2025

Better Neural Network Expressivity: Subdividing the Simplex

This work studies the expressivity of ReLU neural networks with a focus on their depth. A sequence of previous works showed that $\lceil \log_2(n+1) \rceil$ hidden layers are sufficient to compute all continuous piecewise linear (CPWL) functions on $\mathbb{R}^n$. Hertrich, Basu, Di Summa, and Skutella (NeurIPS'21 / SIDMA'23) conjectured that this result is optimal in the sense that there are CPWL functions on $\mathbb{R}^n$, like the maximum function, that require this depth. We disprove the conjecture and show that $\lceil\log_3(n-1)\rceil+1$ hidden layers are sufficient to compute all CPWL functions on $\mathbb{R}^n$. A key step in the proof is that ReLU neural networks with two hidden layers can exactly represent the maximum function of five inputs. More generally, we show that $\lceil\log_3(n-2)\rceil+1$ hidden layers are sufficient to compute the maximum of $n\geq 4$ numbers. Our constructions almost match the $\lceil\log_3(n)\rceil$ lower bound of Averkov, Hojny, and Merkert (ICLR'25) in the special case of ReLU networks with weights that are decimal fractions. The constructions have a geometric interpretation via polyhedral subdivisions of the simplex into ``easier'' polytopes.

cs.LG↗

arXiv2025

On the Depth of Monotone ReLU Neural Networks and ICNNs

We study two models of ReLU neural networks: monotone networks (ReLU$^+$) and input convex neural networks (ICNN). Our focus is on expressivity, mostly in terms of depth, and we prove the following lower bounds. For the maximum function MAX$_n$ computing the maximum of $n$ real numbers, we show that ReLU$^+$ networks cannot compute MAX$_n$, or even approximate it. We prove a sharp $n$ lower bound on the ICNN depth complexity of MAX$_n$. We also prove depth separations between ReLU networks and ICNNs; for every $k$, there is a depth-2 ReLU network of size $O(k^2)$ that cannot be simulated by a depth-$k$ ICNN. The proofs are based on deep connections between neural networks and polyhedral geometry, and also use isoperimetric properties of triangulations.

cs.LG↗

arXiv2023

Linear Families of Triangles and Orthology

Two triangles are called orthologic if the perpendiculars from the vertices of one of them to the sides of the other are concurrent. In this paper, we explore the concept of orthology from various points of view. Mostly we work in terms of elementary geometry in $\mathbb{R}^2$. In the final section, we relate the discussed concepts to $\mathbb{R}^4$. A key idea in this paper is that two orthologic triangles generate a one-parameter (linear) family in which any two triangles are orthologic. Working with such a family can provide a natural approach to certain questions.

math.MG↗