The Sharp Hexagonal Constant in Steinerberger's Geometric Uncertainty Principle on the Square

For a partition of [0,1]² into N measurable pieces Ω_i, let F = Σ_i |Ω_i| A(Ω_i) + Σ_i (|Ω_i| − min_j |Ω_j|), where A is the Fraenkel asymmetry with respect to discs. Steinerberger proved F ≥ 1/60000 for large N and asked, in Oberwolfach Report 49/2013, whether the extremal configuration is a hexagonal tiling, which would give the optimal constant c ≈ 0.07. We show that this is so on the square, in an asymptotic sense, with the constant c_H = 0.07446575455…, the asymmetry of the regular hexagon. Every partition of [0,1]², into any number of pieces, satisfies F > c_H + 0.06 m, where m is the smallest measure of a piece, while for every N a honeycomb partition with a boundary correction has F ≤ c_H + O(N^(−1/2)). Hence the infimum I_N over partitions into N pieces lies between c_H + 0.03/N and c_H + O(N^(−1/2)), I_N tends to c_H as N → ∞, and the value c_H is attained by no partition. We do not show that near-minimisers are geometrically close to a honeycomb, and we do not treat other domains. The proof bounds F below by a covering functional of discs of unequal sizes and uses Laguerre cells, a moment lemma of Fejes Tóth for the indicator of the complement of a disc (proved in an appendix), Euler's formula, and a tangent inequality in the number of sides with explicit constants. The same proof answers, in the same asymptotic sense, a weighted version asked by Burchard: the limit of the infimum is min{α c_H, 1}, so honeycomb partitions stop being asymptotically extremal exactly at the weight α = 1/c_H = 13.4289…. This is an unrefereed note.