<< where f(n) is the geometric mean of chord-lengths of all chords of the unit circle containing say the point 1 >> This is not meaningful without specifying a distribution: is the integration wrt angle, rather than (say) arc length, or some other, more obscure weight function? See https://en.wikipedia.org/wiki/Bertrand_paradox_(probability) WFL On 8/21/18, Dan Asimov <dasimov@earthlink.net> wrote:
Aha: Choose n in Z+ and for each fixed k in 0 <= k < n find the product of the complex numbers
P(k; n) = (zeta_n)^j - (zeta_n)^k, j in {0, 1, ..., n-1} - {k}
over the range 0 <= k < n.
Let Q = Product over k=0,1,..., n-1 of P(k: n),
where zeta_n denotes exp(2*π*i/n).
Then clearly |Q| = (f(n))^n(n-1)
where f(n) is the geometric mean of chord-lengths of all chords of the unit circle containing say the point 1.
The rest of the proof is left as an exercise.
—Dan
----- Thanks everyone, Tom, Warut, James.
The significance of those integrals is that their summing to zero implies:
[The geometric mean of the chord-lengths of all the chords containing some given point of the unit circle] = 1.
Is there maybe a direct way to see that? -----
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