Upper bound on the eventual strictness threshold n0 in the \xi_n–\theta_n inequality

Improve the current upper bound on n0, the minimal natural number such that for all n ≥ n0 the inequality \xi_n(Q_n) < \frac{n+1}{2}(\theta_n(Q_n) - 1) + 1 holds. The best available bounds are 8 ≤ n0 ≤ 53; refine the upper bound and, ideally, determine the exact value of n0.

Background

The paper defines n0 as the minimal dimension beyond which the inequality linking \xi_n(Q_n) and \theta_n(Q_n) is always strict. Currently, estimates give 8 ≤ n0 ≤ 53 based on analytical bounds and computational evidence.

Tightening the upper bound (and finding the exact n0) would clarify when asymptotic phenomena dominate the relationship between minimal absorption and minimal projector norms on cubes.

References

Let $n_0$ be the minimal natural number such that for all $n\geq n_0$ inequality (\ref{nev_strict}) holds. Note that a better estimate from above for $n_0$ is an open problem.

— Geometric Estimates in Linear Interpolation on a Cube and a Ball (2402.11611 - Nevskii, 18 Feb 2024) in Section 4