By a random maximal collection of intervals of length L in [0,1], I mean: First, pick a point at random from the uniform distribution on [0,1] repeatedly until an interval of length L with that point as midpoint fits in [0,1]. That's s_1. (*) Inductively having picked disjoint intervals s_1,...s_n of length L, then if no complementary interval of length > L remains, stop. Then n*L = |I(L)| below. Otherwise, pick a point at random from the uniform distribution on [0,1] - (s_1 + . . . + s_n) repeatedly until it's the midpoint of an interval of length L in that complementary region (so is disjoint from the previous intervals). Now go to (*). --Dan << 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 [Renyi's] 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) . . . I 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).)

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Daniel Asimov Visiting Scholar Department of Mathematics University of California Berkeley, California