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Packing strips in the hyperbolic plane. (English) Zbl 1032.52007

Summary: A strip of radius \(r\) in the hyperbolic plane is the set of points within distance \(r\) of a given geodesic. We define the density of a packing of strips of radius \(r\) and prove that this density cannot exceed \[ {\mathcal S}(r)= {3\over\pi} \sinh r \text{arccosh} \left (1+ {1 \over 2\sinh^2r} \right). \] This bound is sharp for every value of \(r\) and provides sharp bounds on collaring theorems for simple geodesics on surfaces.

MSC:

52C15 Packing and covering in \(2\) dimensions (aspects of discrete geometry)
51M09 Elementary problems in hyperbolic and elliptic geometries
51M04 Elementary problems in Euclidean geometries
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