Global minimizers of the harmonic-cubic Chebyshev ratio

Prove that \(\kappa_3=35/9\) and \(\kappa_4=32/9\), with minimizers, up to orthogonal transformations and scaling, given respectively by \(x_1x_2x_3\) and the Lawson cubic; determine whether \(\kappa_n\) is attained by a Hsiang cubic in every dimension and, if so, by the Hsiang cubic maximizing the Killing trace in that dimension.

Background

The Chebyshev ratio is R(p)=∥p∥∞2/∥p∥22R(p)=\|p\|_\infty^2/\|p\|_2^2 on harmonic cubic forms, and κn\kappa_n is its minimum in dimension nn. The paper proves the exact global minimum in dimensions $2,5,8,14,26$, with Cartan cubics providing the minimizers in the latter four dimensions.

In dimensions three and four, the paper establishes only local minimality for the cubic x1x2x3x_1x_2x_3 and the Lawson cubic. The broader conjecture asks whether all global minimizers arise among Hsiang cubics and whether maximizing the Killing trace identifies the minimizer.

References

Prove \Cref{conj:kappa}. Is $\kappa_n$ attained by a Hsiang cubic for every $n$? If so, by the one maximizing $\trace K$ among the Hsiang cubics in $\R n$?

— Hsiang cubics as spherical Chebyshev polynomials: equioscillation and idempotent quadrature  (2610.13532 - Tkachev, 9 Oct 2026) in Section 7, Problem \ref{prob:global}; Conjecture \ref{conj:kappa} in Section 4.3

In $\R4$, on the other hand, the Lawson cubic ($Pol(2,2)$) gives $\frac19>\frac1{10}$, and the case $n=4$ remains open.

— Hsiang cubics as spherical Chebyshev polynomials: equioscillation and idempotent quadrature  (2610.13532 - Tkachev, 9 Oct 2026) in Section 5, Remark \ref{rem:akuconj}