Fair enough. Let ch_t denote the chord [1, e^it] from the point 1 in C = R^2 to the point e^it. Then for any arc A of the unit circle of length a: let the measure of the set of chords {ch_t | t in the arc A} be meas({ch_t | t in the arc A}) = length(A) / 2π. i.e., just the uniform distribution on the endpoints {e^it}. —Dan ----- << 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.