2/3 power law in the plane - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T22:36:29Z http://mathoverflow.net/feeds/question/85787 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85787/2-3-power-law-in-the-plane 2/3 power law in the plane John Gunnar Carlsson 2012-01-16T04:25:15Z 2012-01-16T22:05:44Z <p>I've recently come across a particular problem whose solution turns out to be a probability distribution given by $f(x) = \alpha \|x\|^{-2/3}$ in the unit disk in $\mathbb{R}^2$ and zero elsewhere (I alluded to this in a previous question</p> <p><a href="http://mathoverflow.net/questions/85624/fitting-a-mesh-to-a-density-function" rel="nofollow">http://mathoverflow.net/questions/85624/fitting-a-mesh-to-a-density-function</a></p> <p>which was very helpfully answered by Anton Petrunin). Does this distribution appear in any other contexts? I've seen a $2/3$ power law in reference to metabolic rates of animals:</p> <p><a href="http://www.ncbi.nlm.nih.gov/pubmed/19906667" rel="nofollow">http://www.ncbi.nlm.nih.gov/pubmed/19906667</a></p> <p>and in kinematics:</p> <p><a href="http://www.ncbi.nlm.nih.gov/pubmed/9844558" rel="nofollow">http://www.ncbi.nlm.nih.gov/pubmed/9844558</a></p> <p>but both of the preceding cases appear to be looking at rather one-dimensional quantities (and they're positive powers rather than negative in my case, not an important distinction); they have $f(t) = \alpha t^{2/3}$, where $t$ represents mass in the first case and angular velocity of the tip of a pen in the second. This seems different from the situation that I'm describing. To put it succinctly,</p> <p>"Are there natural quantities that are proportional to the distance to some point, raised to the $-2/3$ power?"</p> <p>(This may be more appropriate for another forum; if so, I welcome any suggestions)</p> http://mathoverflow.net/questions/85787/2-3-power-law-in-the-plane/85853#85853 Answer by Igor Rivin for 2/3 power law in the plane Igor Rivin 2012-01-16T22:05:44Z 2012-01-16T22:05:44Z <p>It is a theorem of Renyi and Soulanke that the cardinality of the boundary of a convex hull of a uniformly distributed random point set of cardinality $N$ in a smooth convex set grows like $N^{1/3},$ so in particular, if you take a point set in a disk of radius $R,$ so that the density is $1,$ then the cardinality of said convex hull boundary grows like $R^{2/3}.$ A similar statement was shown by Baranyi et al for lattice points in that same disk, see the recent question: <a href="http://mathoverflow.net/questions/85368/convex-hull-of-lattice-points-in-a-circle" rel="nofollow">http://mathoverflow.net/questions/85368/convex-hull-of-lattice-points-in-a-circle</a></p>