Two-dimensional collapse for broader projective functionals

Determine whether the two-dimensional collapse mechanism established for the four-point ratio functional F_d(x,y;u,v) = ((x · u)(y · v))/((x · v)(y · u)) extends to other scale-invariant (projective) functionals defined on positive vectors or matrices, by proving that their worst-case values in arbitrary dimension are realized by two-dimensional supports with the same extremal coordinate-slope ratios.

Background

The paper develops a distortion framework for positive matrices, showing that worst-case misalignment and sharp bounds on distortion occur already in dimension two. A key ingredient is a collapse principle: by pushing coordinate ratios to extremal values and aggregating them, one reduces the analysis to two-dimensional vectors without loss in the worst-case distortion value.

This reduction relies only on the four-point ratio functional F_d(x,y;u,v) = ((x·u)(y·v))/((x·v)(y·u)), which captures distortion in terms of dot products of selected rows and columns. Motivated by this, the authors conjecture that similar two-dimensional collapse mechanisms may hold for other projective (scale-invariant) quantities beyond this specific ratio, potentially broadening the scope of the dimension-collapse method.

References

A second direction concerns other projective quantities. Since the collapse argument in Section~\ref{sec:reduction} uses only the four-point ratios \frac{(x!\cdot!u)(y!\cdot!v)}{(x!\cdot!v)(y!\cdot!u)}, it is natural to conjecture that analogous two-dimensional collapse mechanisms may apply to a broader class of projective functionals.

— Two-Dimensional Projective Collapse and Sharp Distortion Bounds for Products of Positive Matrices  (2512.10872 - Kritchevski, 11 Dec 2025) in Section 6 (Discussion and outlook)