While not wishing to be tiresome, I nevertheless feel constrained to point out that Dan's original statement again begs the question of what exactly is being "randomised". While the parking problem is surely the most natural interpretation, one might instead for instance consider a (scaled-up) process placing unit-length intervals sequentially, separated by gaps chosen uniformly in the unit interval. In this case the mean separation is obviously 1/2, and the limiting density 2/3. Any problem which imprecisely combines geometry and probability is liable to give rise to such Bertrand-style paradoxes (more properly, inconsistencies). WFL On 12/3/12, Veit Elser <ve10@cornell.edu> wrote:
Renyi's parking problem: http://mathworld.wolfram.com/RenyisParkingConstants.html
On Dec 2, 2012, at 8:05 PM, Dan Asimov <dasimov@earthlink.net> wrote:
1) For small L > 0, Let I(L) be a random maximal collection of disjoint closed intervals each of length L and lying in [0,1] in R.
Let |I(L)| denote the total length of all the intervals of I(L).
It seems clear that there exists some constant C(1) such that, with probability 1, the limit of |I(L)| as L -> 0 = C.
Question: Find C. -----------------
One 2D version of this is:
2) For small L > 0 let D(L) denote a random maximal collection of disjoint
closed geometric disks each of diameter L and lying in [0,1]^2 in R^2.
Likewise, as L -> 0 what is the limit C(2) of the total area of the disks of D(L) ? -----------------
n) The nD version: In [0,1]^n, what is C(n) for any n ? -----------------
(If you don't like edge effects, these questions can be asked for the cubical n-torus R^n / Z^n instead of [0,1]^n, with the same answer C(n).)
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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