arXiv subjects

Mikhail Nevskii

Publications and source records attributed to Mikhail Nevskii.

13 recordsLinked to original sources

arXiv2025

On a Simplex Inscribed in a Ball

Let $B_n$ be the $n$-dimensional unit ball given by the inequality $\|x\|\leq 1$, where $\|x\|$ is the standard Euclid norm in ${\mathbb R}^n$. For an $n$-dimensional nondegenerate simplex $S$, we denote by $E$ the ellipsoid of minimum volume which contains $S$. Suppose $S\subset B_n$, $0\leq m\leq n-1$. Let $G$ be any $m$-dimensional face of $S$ and let $H$ be the opposite $(n-m-1)$-dimensional face. Denote by $g$ and $h$ the centers of gravity of $G$ and $H$ respectively. Define $y$ as the intersection point of the line passing from $g$ to $h$ with the boundary of $E$. Let us call the face $G$ suitable if $y\in B_n.$ Earlier it was proved that each simplex $S\subset B_n$ has a suitable face of any dimension $\leq n-1$. We show the following. Let $S$ be inscribed in $B_n$. If some vertex of $S$ is suitable, then there exists a suitable face of any dimension $\leq n-1$ which contains this vertex.

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arXiv2024

Optimal Lagrange Interpolation Projectors and Legendre Polynomials

Let $K$ be a convex body in ${\mathbb R}^n$, and let $\Pi_1({\mathbb R}^n)$ be the space of polynomials in $n$ variables of degree at most $1$. Given an $(n+1)$-element set $Y\subset K$ in general position, we let $P_Y$ denote the Lagrange interpolation projector $P_Y: C(K)\to \Pi_1({\mathbb R}^n)$ with nodes in $Y$. In this paper, we study upper and lower bounds for the norm of the optimal Lagrange interpolation projector, i.e., the projector with minimal operator norm where the minimum is taken over all $(n+1)$-element sets of interpolation nodes in $K$. We denote this minimal norm by $\theta_n(K)$. Our main result, Theorem 5.2, provides an explicit lower bound for the constant $\theta_n(K)$ for an arbitrary convex body $K\subset{\mathbb R}^n$ and an arbitrary $n\ge 1$. We prove that $\theta_n(K)\ge \chi_n^{-1}\left({{\rm vol}(K)}/{{\rm simp}(K)}\right)$ where $\chi_n$ is the Legendre polynomial of degree $n$ and ${\rm simp}(K)$ is the maximum volume of a simplex contained in $K$. The proof of this result relies on a geometric characterization of the Legendre polynomials in terms of the volumes of certain convex polyhedra. More specifically, we show that for every $\gamma\ge 1$ the volume of the set $\left\{x=(x_1,...,x_n)\in{\mathbb R}^n : \sum |x_j| +\left|1- \sum x_j\right|\le\gamma\right\}$ is equal to ${\chi_n(\gamma)}/{n!}$. If $K$ is an $n$-dimensional ball, this approach leads us to the equivalence $\theta_n(K) \asymp\sqrt{n}$ which is complemented by the exact formula for $\theta_n(K)$. If $K$ is an $n$-dimensional cube, we obtain explicit efficient formulae for upper and lower bounds of the constant $\theta_n(K)$; moreover, for small $n$, these estimates enable us to compute the exact values of this constant.

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arXiv2024

Geometric Estimates in Linear Interpolation on a Cube and a Ball

The paper contains a survey of the results obtained by the author in recent years. These results concern the application in multivariate polynomial interpolation of some geometric constructions and methods. In particular, we give estimates of the projector's norms through the characteristics of sets associated with homothety. The known exact values and nowaday best estimates of these norms are given. Also we formulate some open problems. The survey is dedicated to Professor Yuri Brudnyi in connection with the upcoming 90th anniversary of his birth. Contents: Introduction. 1. Notation and preliminaries. 2. The values of $α(Q_n;S)$ and $ξ(Q_n;S)$. 3. The values of $α(B_n;S)$ and $ξ(B_n;S)$. 4. Estimates for $θ_n(Q_n)$. 5. Legendre polynomials and the measure of $E_{n,γ}$. 6. Inequalities $θ_n(Q_n)>c\sqrt{n}$ and $θ_n(B_n)>c\sqrt{n}$. 7. The norm $\|P\|_{B}$ for an inscribed regular simplex. 8. A theorem on a simplex and its minimal ellipsoid. 9. The value of $θ_n(B_n)$. 10. Interpolation by wider polynomial spaces.

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arXiv2023

On a Geometric Approach to the Estimation of Interpolation Projectors

Suppose $Ω$ is a closed bounded subset of ${\mathbb R}^n,$ $S$ is an $n$-dimensional non-degenerate simplex, $ξ(Ω;S):=\min \left\{σ\geq 1: \, Ω\subset σS\right\}$. Here $σS$ is the result of homothety of $S$ with respect to the center of gravity with coefficient $σ$. Let $d\geq n+1,$ $φ_1(x),\ldots,φ_d(x)$ be linearly independent monomials in $n$ variables, $φ_1(x)\equiv 1,$ $φ_2(x)=x_1,\ \ldots, \ φ_{n+1}(x)=x_n.$ Put $Π:={\rm lin}(φ_1,\ldots,φ_d).$ The interpolation projector $P: C(Ω)\to Π$ with a set of nodes $x^{(1)},\ldots, x^{(d)}$ $ \in Ω$ is defined by equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).$ Denote by $\|P\|_Ω$ the norm of $P$ as an operator from $C(Ω)$ to $C(Ω)$. Consider the mapping $T:{\mathbb R}^n\to {\mathbb R}^{d-1}$ of the form $T(x):=(φ_2(x),\ldots,φ_d(x)). $ We have the following inequalities: $ \frac{1}{2}\left(1+\frac{1}{d-1}\right)\left(\|P\|_Ω-1\right)+1$ $ \leq ξ(T(Ω);S)\leq \frac{d}{2}\left(\|P\|_Ω-1\right)+1. $ Here $S$ is the $(d-1)$-dimensional simplex with vertices $T\left(x^{(j)}\right).$ We discuss this and other relations for polynomial interpolation of functions continuous on a segment. The results of numerical analysis are presented.

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arXiv2023

The Minimum Norm of a Projector under Linear Interpolation on a Euclidean Ball

We prove the following proposition. Under linear interpolation on a Euclidean $n$-dimensional ball $B$, an interpolation projector whose nodes coincide with the vertices of a regular simplex inscribed into the boundary sphere has the minimum $C$-norm. This minimum norm $θ_n(B)$ is equal to $\max\{ψ(a_n),ψ(a_n+~1)\}$, where $ψ(t)=\dfrac{2\sqrt{n}}{n+1}\Bigl(t(n+1-t)\Bigr)^{1/2}+ \left|1-\dfrac{2t}{n+1}\right|$, $0\leq t\leq n+1$, and $a_n=\left\lfloor\dfrac{n+1}{2}-\dfrac{\sqrt{n+1}}{2}\right\rfloor$. For any $n$, $\sqrt{n}\leq θ_n(B)\leq \sqrt{n+1}.$ Moreover, $θ_n(B)$ $=$ $\sqrt{n}$ only for $n=1$ and $θ_n(B)=\sqrt{n+1}$ if and only if $\sqrt{n+1}$ is an integer.

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arXiv2022

On Some Estimate for the Norm of an Interpolation Projector

Let $Q_n=[0,1]^n$ be the unit cube in ${\mathbb R}^n$ and let $C(Q_n)$ be a space of continuous functions $f:Q_n\to{\mathbb R}$ with the norm $\|f\|_{C(Q_n)}:=\max_{x\in Q_n}|f(x)|.$ By $Π_1\left({\mathbb R}^n\right)$ denote a set of polynomials of degree $\leq 1$, i.e., a set of linear functions on ${\mathbb R}^n$. The interpolation projector $P:C(Q_n)\to Π_1({\mathbb R}^n)$ with the nodes $x^{(j)}\in Q_n$ is defined by the equalities $Pf\left(x^{(j)}\right)= f\left(x^{(j)}\right)$, $j=1,$ $\ldots,$ $ n+1$. Let $\|P\|_{Q_n}$ be the norm of $P$ as an operator from $C(Q_n)$ to $C(Q_n)$. If $n+1$ is an Hadamard number, then there exists a nondegenerate regular simplex having the vertices at vertices of $Q_n$. We discuss some approaches to get inequalities of the form $||P||_{Q_n}\leq c\sqrt{n}$ for the norm of the corresponding projector $P$.

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arXiv2021

Estimates for Interpolation Projectors and Related Problems in Computational Geometry

This paper contains a survey of results obtained by the authors mostly during the past few years and published by 2021. In particular, we present the best of known estimates of numerical characteristics related to the research theme. Sections: 1. Introduction. 2. The case when $n+1$ is an Hadamard number. 3. Estimates for the minimal absorption index of a cube by a simplex. 4. Estimates for the minimal norm of a projector in linear interpolation on a cube in ${\mathbb R}^n$. 5. Estimates of numbers $ξ_n^\prime$ and $θ_n^\prime$. 6. Simplices satisfying the inclusions $S\subset Q_n\subset nS$. 7. Perfect simplices. 8. Equisecting simplices. 9. Properties of $(0,1)$-matrices of order $n$ having maximal determinant. 10. Problems for a simplex and a Euclidean ball. 11. Linear interpolation on a Euclidean ball. Bibliography: 56 titles. Keywords: simplex, cube, Euclidean ball, homothety, axial diameter, absorption index, Hadamard number, interpolation, projector, norm, estimate.

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arXiv2021

On Properties of a Regular Simplex Inscribed into a Ball

Let $B$ be a Euclidean ball in ${\mathbb R}^n$ and let $C(B)$ be a space of~continuous functions $f:B\to{\mathbb R}$ with the uniform norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|.$ By $Π_1\left({\mathbb R}^n\right)$ we mean a set of polynomials of degree $\leq 1$, i.e., a set of linear functions upon ${\mathbb R}^n$. The interpolation projector $P:C(B)\to Π_1({\mathbb R}^n)$ with the nodes $x^{(j)}\in B$ is defined by the equalities $Pf\left(x^{(j)}\right)= f\left(x^{(j)}\right)$, $j=1,$ $\ldots,$ $ n+1$. The norm of $P$ as an operator from $C(B)$ to $C(B)$ can be calculated by the formula $\|P\|_B=\max_{x\in B}\sum |λ_j(x)|.$ Here $λ_j$ are the basic Lagrange polynomials corresponding to the $n$-dimensional nondegenerate simplex $S$ with the vertices $x^{(j)}$. Let $P^\prime$ be a projector having the nodes in the vertices \linebreak of a regular simplex inscribed into the ball. We describe the points $y\in B$ with the property $\|P^\prime\|_B=\sum |λ_j(y)|$. Also we formulate a geometric conjecture which implies that $\|P^\prime\|_B$ is equal to the minimal norm of an interpolation projector with nodes in $B$. We prove that this conjecture holds true at least for $n=1,2,3,4$. Keywords: regular simplex, ball, linear interpolation, projector, norm

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arXiv2019

Geometric Estimates in Interpolation by Linear Functions on a Euclidean Ball

Let $B_n$ be the Euclidean unit ball in ${\mathbb R}^n$ given by the inequality $\|x\|\leq 1$, $\|x\|:=\left(\sum\limits_{i=1}^n x_i^2\right)^{\frac{1}{2}}$. By $C(B_n)$ we mean the space of continuous functions $f:B_n\to{\mathbb R}$ with the norm $\|f\|_{C(B_n)} := \max\limits_{x\in B_n}|f(x)|$. The symbol $Π_1\left({\mathbb R}^n\right)$ denotes the set of polynomials in $n$ variables of degree $\leq 1$, i.e., the set of linear functions upon ${\mathbb R}^n$. Assume $x^{(1)}, \ldots, x^{(n+1)}$ are the vertices of an $n$-dimensional nondegenerate simplex $S\subset B_n$. The interpolation projector $P:C(B_n)\to Π_1({\mathbb R}^n)$ corresponding to $S$ is defined by the equalities $Pf\left(x^{(j)}\right) = f\left(x^{(j)}\right).$ Denote by $\|P\|_{B_n}$ the norm of $P$ as an operator from $C(B_n)$ onto $C(B_n)$. We describe the approach in which $\|P\|_{B_n}$ can be estimated from below via the volume of $S$.

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arXiv2019

Interpolation by Linear Functions on an $n$-Dimensional Ball

By $B=B(x^{(0)};R)$ we denote the Euclidean ball in ${\mathbb R}^n$ given by the inequality $\|x-x^{(0)}\|\leq R$. Here $x^{(0)}\in{\mathbb R}^n, R>0$, $\|x\|:=\left(\sum_{i=1}^n x_i^2\right)^{1/2}$. We mean by $C(B)$ the space of continuous functions $f:B\to{\mathbb R}$ with the norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|$ and by $Π_1\left({\mathbb R}^n\right)$ the set of polynomials in $n$ variables of degree $\leq 1$, i.e., linear functions on ${\mathbb R}^n$. Let $x^{(1)}, \ldots, x^{(n+1)}$ be the vertices of $n$-dimensional nondegenerate simplex $S\subset B$. The interpolation projector $P:C(B)\to Π_1({\mathbb R}^n)$ corresponding to $S$ is defined by the equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).$ We obtain the formula to compute the norm of $P$ as an operator from $C(B)$ into $C(B)$ via $x^{(0)}$, $R$ and coefficients of basic Lagrange polynomials of $S$. In more details we study the case when $S$ is a regular simplex inscribed into $B_n=B(0,1)$.

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arXiv2019

On Some Problems Related to a Simplex and a Ball

Let $C$ be a convex body and let $S$ be a nondegenerate simplex in ${\mathbb R}^n$. Denote by $ξ(C;S)$ the minimal $τ>0$ such that $C$ is a subset of the simplex $τS$. By $α(C;S)$ we mean the minimal $τ>0$ such that $C$ is contained in a translate of $τS$. Earlier the author has proved the equalities $ξ(C;S)=(n+1)\max\limits_{1\leq j\leq n+1} \max\limits_{x\in C}(-λ_j(x))+1$ \ (if $C\not\subset S$), \ $α(C;S)= \sum\limits_{j=1}^{n+1} \max\limits_{x\in C} (-λ_j(x))+1.$ Here $λ_j$ are linear functions called the basic Lagrange polynomials corresponding to $S$. In his previous papers, the author has investigated these formulae if $C=[0,1]^n$. The present paper is related to the case when $C$ coincides with the unit Euclidean ball $B_n=\{x: \|x\|\leq 1\},$ where $\|x\|=\left(\sum\limits_{i=1}^n x_i^2 \right)^{1/2}.$ We establish various relations for $ξ(B_n;S)$ and $α(B_n;S)$, as well as we give their geometric interpretation.

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arXiv2018

Properties of 0/1-Matrices of Order n Having Maximum Determinant

We give some necessary conditions for maximality of $0/1$-determinant. Let ${\bf M}$ be a nondegenerate $0/1$-matrix of order $n$. Denote by $\bf A$ the matrix of order $n+1$ which appears from ${\bf M}$ after adding the $(n+1)$th row $(0,0,\ldots,0,1)$ and the $(n+1)$th column consisting of $1$'s. Suppose ${\bf A}^{-1}=(l_{ij}),$ then for all $i=1,\ldots,n$ we have $\sum_{j=1}^{n+1} |l_{ij}|\geq 2.$ Moreover, if $|\det({\bf M})|$ is equal to the maximum value of a $0/1$-determinant of order $n$, then $\sum_{j=1}^{n+1} |l_{ij}|= 2$ for all $i=1,\ldots,n$. Keywords: maximum 0/1-deteminant, simplex, cube, axial diameter

math.MG↗

arXiv2017

Five-dimensional Perfect Simplices

Let $Q_n=[0,1]^n$ be the unit cube in ${\mathbb R}^n$, $n \in {\mathbb N}$. For a nondegenerate simplex $S\subset{\mathbb R}^n$, consider the value $ξ(S)=\min \{σ>0: Q_n\subset σS\}$. Here $σS$ is a homothetic image of $S$ with homothety center at the center of gravity of $S$ and coefficient of homothety $σ$. Let us introduce the value $ξ_n=\min \{ξ(S): S\subset Q_n\}$. We call $S$ a perfect simplex if $S\subset Q_n$ and $Q_n$ is inscribed into the simplex $ξ_n S$. It is known that such simplices exist for $n=1$ and $n=3$. The exact values of $ξ_n$ are known for $n=2$ and in the case when there exist an Hadamard matrix of order $n+1$, in the latter situation $ξ_n=n$. In this paper we show that $ξ_5=5$ and $ξ_9=9$. We also describe infinite families of simplices $S\subset Q_n$ such that $ξ(S)=ξ_n$ for $n=5,7,9$. The main result of the paper is the existence of perfect simplices in ${\mathbb R}^5$. Keywords: simplex, cube, homothety, axial diameter, Hadamard matrix

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