Optimal densities of packings consisting of highly unequal objects

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Let Δ be the optimal packing density of 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 ΔD + (1 - ΔD) Δ{B} as r tends to zero.