Here’s an interesting article about this in R^n. http://www3.math.tu-berlin.de/combi/wp_henk/wp-content/uploads/2015/03/hyper... On Sun, Oct 8, 2017 at 15:20 Dan Asimov <dasimov@earthlink.net> wrote:

Another pretty lowbrow way to define a probability measure on the planes meeting a cube is to first pick a random point in the cube, then pick a random direction through that point and take the perpendicular plane. Where random means uniformly distributed.

Maybe like the Bertrand paradox (about a random chord of the circle) there are multiple "reasonable" probability measures to choose from, with inequivalent measures?

—Dan

Jim Propp wrote: ----- If I choose a random plane that intersects a cube, creating a polygon, how many sides does the polygon have on average?

(The most lowbrow way to define the probability measure I have in mind when I say "random plane" is to circumscribe a sphere of radius r around the cube with center O. Choose a point uniformly P at random on the sphere, and now choose a point Q uniformly at random on segment OP. If Q is in the cube, erect a plane through Q perpendicular to OP. Otherwise, start over. The most highbrow way is to go via the standard measure on the affine Grassmannian of all planes in 3-space; if we restrict to the set of planes that pass through the cube, we get a finite measure, so we can rescale to get a probability measure.) -----

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