The Gelfand widths of $\ell_p$-balls for $0<p\leq 1$

TL;DR

This paper establishes precise bounds for Gelfand widths of $ ext{ell}_p$-balls in finite-dimensional spaces, advancing understanding in compressive sensing theory.

Contribution

It provides the first sharp bounds for Gelfand widths of $ ext{ell}_p$-balls for $0<p extless 1$, connecting geometric functional analysis with compressive sensing.

Findings

Sharp lower and upper bounds for Gelfand widths are derived.

Results are applicable to the theory of compressive sensing.

The bounds improve previous estimates in the literature.

Abstract

We provide sharp lower and upper bounds for the Gelfand widths of $\ell_p$-balls in the $N$-dimensional $\ell_q^N$-space for $0<p\leq 1$ and $p<q \leq 2$. Such estimates are highly relevant to the novel theory of compressive sensing, and our proofs rely on methods from this area.