Establish the column-combining condition for all small-norm projections

Prove that every projection P=(p_{ij}) from \ell_\infty^N onto an arbitrary subspace X\subset\ell_\infty^N, with \|P\|\leq 1+\varepsilon<4/3, admits pairwise disjoint nonempty sets A_1,\ldots,A_n\subset[1:N] and indices k_i\in A_i, where n=\dim X, such that \sum_{j\in A_i}|p_{k_i j}|\geq 1-\varepsilon for every i\in[1:n].

Background

The proof of the main Banach–Mazur estimate reduces the problem to a column-combining condition for a projection onto an n-dimensional subspace X of \ell_\inftyN. For codimension-two subspaces, the paper establishes this condition in Theorem \ref{base}; for hyperplanes, an analogous result follows from Proposition \ref{hypergeom}.

The authors construct higher-codimension examples showing that the geometric argument used for codimension two cannot be directly extended. Nevertheless, those examples still satisfy the column-combining condition, and the authors report that they have found no counterexample among projections of norm less than 4/3. Proving the condition in general would yield the conjectured Banach–Mazur estimate without a codimension restriction.

References

So far, we have not found a counterexample to colcomb among projections of norm less than $4/3$, in any codimension. It is therefore natural to ask whether this condition holds for every such projection onto a subspace of $\ell_\inftyN$.

colcomb:

∑j∈Ai∣pkij∣≥1−ε,i∈[1:n].\sum_{j\in A_i}|p_{k_i j}|\geq1-\varepsilon, \qquad i\in[1:n].

— Projection Constants and Banach-Mazur Distance in Codimension Two  (2610.06141 - Deregowska et al., 5 Oct 2026) in Section 3, Discussion, final paragraphs