Timeline for Width of a random convex polygon
Current License: CC BY-SA 3.0
10 events
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| Jun 5, 2017 at 13:36 | comment | added | Joseph O'Rourke | Recent related paper: Kabluchko, Zakhar, Vladislav Vysotsky, and Dmitry Zaporozhets. "Convex hulls of random walks, hyperplane arrangements, and Weyl chambers." arXiv:1510.04073 (2015). | |
| Oct 6, 2013 at 14:00 | comment | added | toni | A related and perhaps easier question might be the following: What is the distribution of the diameter of the smallest sphere entirely containing the random walk. This question was discussed by Weiss and Rubin here in 1983. Is it possible that the question is still open? I tried combing the literature, but could not find anything. Perhaps I have missed some very important works. | |
| Oct 6, 2013 at 13:37 | comment | added | toni | In 1941 Daniels was the first to investigate the extent or span of a 1D random walk. He actually determined the probability density of the span of a random walk in 1D. G.H. Weiss and R.J. Rubin generalized the notion of span to multi-dimensional random walks. Spans are the sides of the smallest rectangular box with sides parallel to the coordinate axes that entirely contains the random walk. In 2D, they were able to determine the probability densities for the smallest and largest spans. This is as close as I could get to the answer. | |
| Oct 6, 2013 at 10:06 | comment | added | j.c. | You're looking for information on the convex hull of a random walk on which there is some literature. In particular, this paper of Spitzer and Widom ams.org/journals/proc/1961-012-03/S0002-9939-1961-0130616-7 computes the expected perimeter of such a polygon, using Cauchy's surface area formula which relates the perimeter to the mean width (averaged over projections onto lines in all directions). Unfortunately, I don't (yet?) see how to compute from their work anything about the minimum width (over all projections). | |
| Oct 6, 2013 at 9:53 | comment | added | j.c. | @JosephO'Rourke your link is about computing the average width (usually known as the "mean width") rather than what is wanted in this question, the (minimum) width. | |
| Oct 6, 2013 at 0:18 | answer | added | Joseph O'Rourke | timeline score: 7 | |
| Oct 5, 2013 at 22:16 | history | edited | Ricardo Andrade |
replaced deprecated tag 'geometry'
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| Oct 5, 2013 at 22:04 | comment | added | Joseph O'Rourke | There is info available, not necessarily answering your precise question, e.g. What's the average width of a convex polygon?. | |
| Oct 5, 2013 at 21:58 | review | First posts | |||
| Oct 5, 2013 at 22:26 | |||||
| Oct 5, 2013 at 21:43 | history | asked | toni | CC BY-SA 3.0 |