Interesting question! Which also suggests some related questions: I) -- What about the expected intersection of two hemi-n-spheres in the n-sphere S^n ? In S^0 the average intersection has 0-area equal to 1/4 that of S^0. In S^1 the average intersection has 1-area equal to 1/4 that of S^1. In S^2 the average intersection has 2-area equal to ... ...Is it again 1/4 that of S^2 ??? (Dubious argument: Two random hemispheres are determined by two random *oriented* 2-planes through the origin of R^3 (or equivalently, two random unit vectors. The (least) angle between the planes is the same as that between the vectors, so is anywhere from 0 to 180 degrees, averaging 90 degrees. Two hemispheres at 90 degrees intersect in half a hemisphere, again giving 1/4 the n-area of S^n, this time for n = 2. Hey, I said it was dubious.) II) --- One could ask the analogous question to Jim's: What is the expected (n-1)-area of the intersection of two random hemispheres of S^n ??? III) ---- Another nice space to ask both the I) and II) (Jim's) types of questions about is the torus. The simplest torus in all dimensions is the cubical one, T^n = R^n / Z^n obtained from a unit cube by identifying corresponding points on opposite faces. The only "drawback" on T^n is that perhaps there is no obvious radius of a ball or sphere to ask about. So maybe here it's appropriate to ask, for any two given radii r > 0 and s > 0, about the expected intersection n-area (or boundary (n-1)-area) of two random balls with those radii. III) ---- Something I worked on a long time ago is the n-dimensional version(s) of this question*: Given two geometric circles (each the intersection of a 2-sphere and a 2-plane through its center) in the 3-torus T^3 of radii r and s, what is the probability that they *link* each other? This turns out to be pretty interesting. —Dan —————————————————————————————————————————————————————————————————— * which requires spheres S^k of radius r and S^m of radius s, with k + m = n - 1 so that linking can take place. Jim Propp wrote: ----- Let P be a point in Euclidean n-space and P' be a point chosen uniformly at random from the ball of radius 2 centered at P, so that the unit balls centered at P and P' have nonempty intersection. Show that the expected hypersurface area of the intersection of the balls (I guess the shape would be called a lunoid) is exactly equal to the hypersurface area of an (n-1)-sphere of radius 1/2. Is this observation new? What about the case n=2 where the lunoid is a lune? I know a proof, but it uses combinatorics in a way that I suspect is unnecessary. -----