The mean diagonal of a random convex polygon of fixed perimeter

Draw a planar n-gon of perimeter one from the symmetric measure, the push-forward of Haar measure on the Grassmannian of 2-planes in ℝn, and condition on convexity. We study Mn, the expected mean length of its n(n − 3)/2 diagonals. For every n the chord means are reduced to integrals of dimension at most three, built from an elementary weight and from gap-void functions that obey a one-step recurrence in the dimension. Consequently πn−2Mn is a period of multiple polylogarithms at fourth roots of unity of weight at most n for even n and at most n − 1 for odd n. A two-dimensional contour integral for the gap-void functions turns the reduction into a representation of Mn by integrals over at most seven variables, in which n is only a parameter. We prove Mn = 2/π2 + (1 + (7/2)ζ(3))/(π2n) + o(1/n). For n ≤ 8 we evaluate Mn in closed form: M4 = 3π2/128 + 9/(8π2) and M5 = 32(2 + log 2)/(15π2) − 4/15 with proofs, M6 and M7 by exact symbolic reduction, and M8, which involves alternating multiple zeta values of weight eight such as ζ(5, 3) and ζ(7̄, 1̄), by numerical identification beyond one hundred digits. We also show that a random quadrilateral whose edge vectors are exchangeable and almost surely in general position is convex, reflex or self-intersecting with probability 1/3 each, and that the symmetric random pentagon is simple with probability 7/24 + (10 − 10 log 2 − (5/2) log2 2)/π2. MSC 2020: 60D05 (primary); 52A22, 11M32, 33B30, 15B52 (secondary) Files: the paper (PDF), its LaTeX source (ZIP) and the programs that reproduce the numerical checks, the figure data and the evaluations of the closed forms, with their input data (ZIP).