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On the number of intersections of two polygons

Jakub Černý, Jan Kára, Daniel Král', Pavel Podbrdský, Miroslava Sotáková, Robert Šámal (2003)

Commentationes Mathematicae Universitatis Carolinae

We study the maximum possible number f ( k , l ) of intersections of the boundaries of a simple k -gon with a simple l -gon in the plane for k , l 3 . To determine the number f ( k , l ) is quite easy and known when k or l is even but still remains open for k and l both odd. We improve (for k l ) the easy upper bound k l - l to k l - k / 6 - l and obtain exact bounds for k = 5 ( f ...

Windings of planar random walks and averaged Dehn function

Bruno Schapira, Robert Young (2011)

Annales de l'I.H.P. Probabilités et statistiques

We prove sharp estimates on the expected number of windings of a simple random walk on the square or triangular lattice. This gives new lower bounds on the averaged Dehn function, which measures the expected area needed to fill a random curve with a disc.

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