Rigorous certification of the high-dimensional numerical threshold

Establish, using rigorous interval arithmetic, the inequality \(\kappa(\Omega_{\alpha,d})>\kappa(\Omega_{\alpha,d}^{\ast})\) for any prescribed fixed pair \((\alpha,d)\) by uniformly bounding \(F_{d-1}(u,s)\) on the region \(\{(u,s):u^{2}+\alpha^{2}s^{2}\leq\kappa(\Omega_{\alpha,d}^{\ast})^{2}\}\).

Background

For dimensions $d\geq3$, the paper analytically proves that sufficiently elongated bipyramids $\Omega_{\alpha,d}$ outperform their equal-volume Euclidean balls, and that the ratio of their Fourier-zero distances diverges as $\alpha\to\infty$. The authors also provide non-rigorous numerical ratios at $\alpha=100$ for several dimensions.

The remark identifies a concrete unresolved computational task: rigorously certifying the comparison for any fixed numerical pair $(\alpha,d)$, rather than relying on the asymptotic theorem or non-rigorous numerical evaluation. The proposed route is interval-arithmetic control of the relevant two-variable function over the specified region.