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How to build a barricade. (English) Zbl 0542.52010
Given a packing \(\mathcal C=\{C_i\mid i\in I\}\) of congruent copies of a strictly convex planar set \(C\) in a parallel strip \(S\) of width \(w\) and a straight line \(\ell\), let \(| \ell \cap \mathcal C|\) denote the sum of the lengths of the segments \(\ell \cap C_i\) over all \(i\in I\). It is proved that \[ \sup_{\mathcal C}\inf_{\ell}| \ell \cap {\mathcal C}| =(\delta(C)+o(1))w, \] where \(\delta (C)\) is the density of the densest packing of congruent copies of \(C\) in the plane. Some generalizations are also considered.
Reviewer: Peter Frankl

MSC:
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
60D05 Geometric probability and stochastic geometry
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References:
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