Estimates for Interpolation Projectors and Related Problems in Computational Geometry

TL;DR

This survey compiles recent results and estimates related to interpolation projectors and geometric problems in computational geometry, focusing on simplices, cubes, and Euclidean balls, with a comprehensive review of known bounds and properties.

Contribution

It provides a consolidated overview of the best known estimates and properties related to interpolation projectors and geometric configurations, highlighting recent advances and open problems.

Findings

Best known estimates for minimal absorption index of a cube by a simplex

Properties of simplices satisfying specific inclusion relations

Maximal determinants of (0,1)-matrices

Abstract

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 $\xi_n^\prime$ and $\theta_n^\prime$. 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.