The average projected length of a unit segment in the plane over orthogonal projections onto lines in all directions is (1/π)(Integral_{0 to π} cos(t) dt = 2/pi. If you approximate a convex closed curve S in the plane by a polygon P, you can sum the above equation over all segments of P to see that the average projected length of S is length(S) / π. This generalizes to higher dimensions. —Dan I wrote: ----- I think Cauchy proved the 2-dimensional version of this by averaging orthogonal projections over all different directions. So there may be a proof like that in higher dimensions. Convex surfaces are almost everywhere C^2, which may help. -----