Volume estimates and their applications in the problem of optimal recovery

TL;DR

This paper develops volume estimates for convex bodies in high-dimensional spaces to derive bounds on Gelfand widths, with applications to Sobolev class approximation in functional analysis.

Contribution

It introduces a novel approach using volume estimates of John-L"owner ellipsoids to evaluate Gelfand and Kolmogorov widths of operators, especially for Sobolev spaces.

Findings

Established lower bounds for section radii of convex bodies.

Derived sharp order estimates for widths of Sobolev classes in $L_q$.

Provided a new method for evaluating widths of multiplier operators.

Abstract

We study volumes of sections of convex origin-symmetric bodies in $\mathbb{R}% ^{n}$ induced by orthonormal systems on probability spaces. The approach is based on volume estimates of \ John-L\"{o}wner ellipsoids and expectations of norms induced by the respective systems. The estimates obtained allows us to establish lower bounds for the radii of sections which gives lower bounds for Gelfand widths (or linear cowidths). As an application we offer a new method of evaluation of Gelfand and Kolmogorov widths of multiplier operators. In particular, we establish sharp orders of widths of standard Sobolev classes $W_{p}^{\gamma }$ in $L_{q}$ in the difficult case, i.e. $% 1<q\leq p\leq \infty $.