<< 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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