Projection constant–Banach–Mazur distance conjecture

For every integer $N\ge 1$ and every linear subspace $V\subset \ell_\infty^N$ with absolute projection constant $\lambda(V)<\frac{4}{3}$ , one has

d\bigl(V,\ell_\infty^{\dim V}\bigr)\le \frac{\lambda(V)}{4-3\lambda(V)}.

References

Primary source

arXiv

Additional references

Projection Constants and Banach-Mazur Distance in Codimension Two — arXiv — Beata Deregowska, Barbara Lewandowska

banach-spaces functional-analysis

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Claimed progress

A new unrefereed preprint proves the conjectured estimate for subspaces of codimension at most two, but the general conjecture remains open.

The conjecture concerns all subspaces and predicts the estimate $d(V,\ell_\infty^{\dim V})\le \lambda(V)/(4-3\lambda(V))$ . No proposer or date is identified in the retrieved material.

Codimension-two result

Beata Deregowska and Barbara Lewandowska claim the estimate for subspaces of codimension one or two. This establishes two concrete low-codimension cases, while the unrestricted conjecture remains open; the preprint is unrefereed.

Current status (as of October 2026): The estimate is claimed for codimension-one and codimension-two subspaces, but the unrestricted conjecture remains open and the claim is unverified.

Sources

No solutions have been posted yet.