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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

Fitting a mesh to a density function

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:

http://www.ncbi.nlm.nih.gov/pubmed/19906667

and in kinematics:

http://www.ncbi.nlm.nih.gov/pubmed/9844558

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,

"Are there natural quantities that are proportional to the distance to some point, raised to the $-2/3$ power?"

(This may be more appropriate for another forum; if so, I welcome any suggestions)

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On the risking of writing a completely useless comment, I'll mention that I find this question very interesting. –  Mark Jan 16 '12 at 12:45
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It seems that the following paper uses such a $-2/3$ approximate density: springerlink.com/content/b70u184653071027 –  Suvrit Jan 16 '12 at 13:30
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Also, this is probably a totally tangential to your question, but this somehow reminds me of Kepler's third law: "The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit," which translates into $P^2 \propto a^3$, or $a \propto P^{2/3}$.... –  Suvrit Jan 16 '12 at 13:37
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Keep your eye out for a power law with exponent $-\frac{5}{3}$, for its integral would yield $-\frac{2}{3}$. $5/3$ power-law dependence shows up, e.g., in wave propagation through turbulent media. –  Joseph O'Rourke Jan 16 '12 at 13:52
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1 Answer

up vote 3 down vote accepted

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: Convex hull of lattice points in a circle

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Good one, thanks! –  John Gunnar Carlsson Jan 17 '12 at 18:22
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