[math-fun] Random nonempty intersection of balls
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. Jim Propp
I found a non-combinatorial proof. The claim applies to all convex bodies, not just balls. Jim Propp On Tuesday, December 19, 2017, James Propp <jamespropp@gmail.com> 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.
Jim Propp
Oops! Veit Elser (in private email) correctly points out that I should've restricted to centrosymmetric convex bodies B. (The proof makes use of the fact that the volume of the Minkowski difference B-B is 2^n times the measure of B, which requires that B is congruent to -B, i.e., that B is centrosymmetric.) Jim Propp On Tue, Dec 19, 2017 at 6:02 PM, James Propp <jamespropp@gmail.com> wrote:
I found a non-combinatorial proof. The claim applies to all convex bodies, not just balls.
Jim Propp
On Tuesday, December 19, 2017, James Propp <jamespropp@gmail.com> 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.
Jim Propp
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James Propp