Journal of Integer Sequences, Vol. 16 (2013), Article 13.2.8

On the Entropy of Curves

Michael Maurice Dodson
University of York
York YO10 5DD
United Kingdom

Michel Mendès France
Université Bordeaux I
351 cours de la Libération
33405 Talence Cedex
F-33405 France


Using geometric probability, we apply the formal definitions of Shannon entropy and Rényi's generalization to study the complexity of planar curves of finite length within a convex set. The bounds for the Shannon and Rényi entropies depend on the arc length of the curve and on that of the boundary of the convex set; they involve a Gibbs distribution and a power law distribution, respectively. We also obtain explicit formulae for the two entropies and determine convex sets that maximize the entropy of curves.

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Received June 28 2012; revised version received August 31 2012. Published in Journal of Integer Sequences, March 2 2013.

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