Optimal densities of packings consisting of highly unequal objects

2016/03/03 by David de Laat, de Laat, David
Mathematics · #52C17 #FOS: Mathematics #Metric Geometry (math.MG) #math.MG #msc:52C17

paper · pdf · doi:10.48550/arxiv.1603.01094

arxiv created 2016/03/03 · arxiv updated 2016/03/04

Abstract

Let Δ be the optimal packing density of \mathbb Rⁿ by unit balls. We show the optimal packing density using two sizes of balls approaches Δ+ (1 - Δ) Δ as the ratio of the radii tends to infinity. More generally, if B is a body and D is a finite set of bodies, then the optimal density Δ_\rB\ ∪ D of packings consisting of congruent copies of the bodies from \rB\ ∪ D converges to Δ_D + (1 - Δ_D) Δ_\B\ as r tends to zero.

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