Work

Widths and rigidity of unconditional sets and random vectors

Year: 2025

Type: article

Abstract: We prove that any unconditional set in $\mathbb{R}^N$ invariant under cyclic shifts of coordinates is rigid in $\ell_q^N$, $1\le q\le 2$, that is, it cannot be well approximated by linear spaces of dimension essentially smaller than $N$. We apply E. D. Gluskin's approach to the setting of averaged Kolmogorov widths of unconditional random vectors or vectors of independent mean zero random variables, and prove their rigidity. These results are obtained using a general lower bound for the averaged Kolmogorov width via weak moments of biorthogonal random vector. This paper continues the study of the rigidity initiated by the first author. We also provide several corollaries including new bounds for Kolmogorov widths of mixed norm balls.

Source: Izvestiya Mathematics

DOI: 10.4213/im9620e

Authors: Yuri Viatcheslavovich Malykhin, Konstantin Sergeevich Ryutin

Institutions: Lomonosov Moscow State University, Steklov Mathematical Institute, Moscow Center of Fundamental and Applied Mathematics

Language: English

Topic: Advanced Banach Space Theory

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: bronze

Funder: Russian Science Foundation

Award: G1125018171