No-dimensional results of combinatorial convexity. Dimension strikes back

TL;DR

This paper explores approximate, dimension-independent versions of classical convexity theorems, discusses open problems, and introduces methods to relate these results to dimension-dependent spaces for practical applications.

Contribution

It introduces no-dimensional convexity results, surveys recent progress, and proposes a way to connect dimension-independent and dimension-dependent frameworks for applications.

Findings

Derived a weak additive analogue of Johnson--Lindenstrauss lemma

Established local-to-global estimates for Chebyshev regression

Provided a local-to-global guarantee for quantum feasibility

Abstract

We discuss no-dimensional (approximate) versions of Carath\'eodory's and Helly's theorems. Our goal is to draw attention to open problems and potential applications related to these results. We survey recent progress and pose several questions. We also point out a simple way to ``bring the dimension back into the picture'': by combining no-dimensional statements with dimension-dependent norm comparisons, one can transfer problems in $\ell_1^d$, $\ell_\infty^d$, and Schatten classes $S_1, S_\infty$ to nearby $\ell_p^d$ or $S_p$ spaces with better geometry. As elementary applications, we obtain a weak additive analogue of the Johnson--Lindenstrauss flattening lemma, local-to-global estimates for Chebyshev regression over the $\ell_1$ ball, and a local-to-global guarantee for quantum feasibility from locally consistent linear measurements.