Theoretical analysis establishes a 1.37^(√d) lower bound for diameter partitions in Euclidean d-space, suggesting tighter constraints on high-dimensional geometric decomposition.
Preprint. Let b(d) be the least number of parts of strictly smaller diameter needed to partition any bounded set of positive diameter in Euclidean d-space. We prove that for every sufficiently large integer d one has b(d) ≥ 1.37 to the power √d. More precisely, the lim inf as d → ∞ of b(d) to the power 1/√d is at least (1249327/1000000) to the power √2, hence greater than 1.37. The elementary argument uses four rational endpoint calculations and exact integer rounding. This record contains the main article, a supplement with exact finite certificates, and a one-page extended abstract. Companion manuscripts: A 1.30^d Lower Bound for the Chromatic Number of Euclidean Space (https://doi.org/10.5281/zenodo.22716236) and the coefficient-rank transfer method for geometric graphs.
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Ilya Hoffman (2026) studied this question.
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