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Classify dimensions where regular inscribed simplices achieve projector norm sqrt(n+1) on the cube

Classify all dimensions n for which there exists at least one n-dimensional regular simplex with vertices among those of the cube Q'_n=[-1,1]^n such that the corresponding linear interpolation projector P:C(Q'_n)→\Pi_1(\mathbb{R}^n) satisfies \|P\|_{Q'_n}=\sqrt{n+1}.

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Background

When n+1 is a Hadamard number, one can inscribe regular simplices with vertices at cube vertices and obtain the upper bound |P|_{Q'_n}≤\sqrt{n+1}. Equality may hold for all such simplices (e.g., n=1,3), for some but not all (e.g., n=15), or for none, depending on n and the chosen Hadamard matrix.

A complete characterization of those n for which equality holds for at least one such simplex remains unknown and impacts sharp upper bounds for minimal projector norms on cubes.

References

The equality $|P|_{Q_n\prime}=\sqrt{n+1}$ may hold as for all regular simplices having vertices at vertices of the cube $(n=1, n=3)$, as for some of them $(n=15),$ or may not be executed at all. The problem of full description of dimensions $n$ with such property is still open.

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