Let v_1,v_2,…,v_n be unit vectors in a real Hilbert space H. Fejes Tóth's real Hilbert-space zone conjecture. There exists a unit vector u∈H such that |langle v_k,u rangle|≥ sin(fracpi2n) for every 1≤ k≤ n. This is presented as a reformulation of Fejes Tóth's equal-width zone conjecture and as an optimal plank problem for real Hilbert spaces. The source does not supply a resolution status for this formulation.

Fejes Tóth's real Hilbert-space zone conjecture

Fejes Tóth posed the underlying equal-width zone conjecture in 1973. The displayed inequality is its vector formulation; the retrieved literature proves the corresponding statement in every finite-dimensional real space.

Known results

The cases of $3$ and $4$ zones were proved by Rosta (1972) and Linhart (1974).

Finite-dimensional resolution (2017–2021)

Jiang and Polyanskii proved that covering a unit sphere by zones requires total width at least $\pi$ , completely resolving the zone conjecture; equal widths yield the bound $\sin\!\left(\frac{\pi}{2n}\right)$ . Ortega-Moreno gave an independent proof for equal-width zones, later simplified by Zhao. Since the finitely many input vectors span a finite-dimensional subspace, this covers the stated Hilbert-space formulation.

Current status (as of September 2026): The finite-dimensional theorem is settled and covers the displayed real Hilbert-space statement; no contrary result was found.

Fejes Tóth's real Hilbert-space zone conjecture

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Progress summary

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