Extremal trilinear forms in dimension three

Determine the sharp constant \(d(3)\) in the inequality \(\|T\|_2^2\le d(3)\|T\|^2\) for real trilinear forms on \((\mathbb R^3)^3\), and classify the trilinear forms attaining it.

Background

The paper relates the trilinear-form ratio ∥T∥2/∥T∥22\|T\|^2/\|T\|_2^2 to the Chebyshev ratio of associated block-trilinear harmonic cubics. For dimension three, the determinant form has Hilbert–Schmidt norm squared six, while the sharp constant is only bounded by 7≤d(3)≤7.367\le d(3)\le7.36.

The extremal trilinear forms are not identified, so the corresponding sharp block-trilinear extremal problem remains unresolved.

References

For $k=3$ the tripling $\det X$ is not extremal for ckp either, since $|T|_22=6<7\le d(3)$; the extremal trilinear forms on $(\R3)3$ are not known.

ckp:

∥T∥22≤d(k) ∥T∥2.\|T\|_2^2\le d(k)\,\|T\|^2 .

— Hsiang cubics as spherical Chebyshev polynomials: equioscillation and idempotent quadrature  (2610.13532 - Tkachev, 9 Oct 2026) in Section 5.1, Remark \ref{rem:tripling}