The union of N essentially distinct δ-tubes has volume at least min{√N δ^{n-1}, N δ^{2n-2}} up to a constant factor, and all near-extremal configurations are characterized.
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On the union of essentially distinct $\delta$-tubes
The union of N essentially distinct δ-tubes has volume at least min{√N δ^{n-1}, N δ^{2n-2}} up to a constant factor, and all near-extremal configurations are characterized.