an orange peel in high dimensions: what fraction of the volume does the peel comprise?

gistfile1.txt

using LinearAlgebra

"""
sample a uniform random point inside a unit orange centered at the origin.

* `d::Int`: dimension of the space
"""
function random_point_inside_unit_orange(d::Int)
  x = 2 * (rand(d) .- 0.5)
  if norm(x) < 1.0
    return x
  else
    return random_point_inside_unit_orange(d)
  end
end

"""
via MC sim, determine the fraction of a unit orange that comprises the peel.

* `δr::Float64`: thickness of the peel
* `d::Int`: dimension of the space
* `nb_samples::Int`: # of MC samples
"""
function fraction_peel(δr::Float64, d::Int;
             nb_samples::Int=10000)
  nb_points_in_peel = 0
  for s = 1:nb_samples
    x = random_point_inside_unit_orange(d)
    if norm(x) > 1.0 - δr
      nb_points_in_peel += 1
    end
  end
  return nb_points_in_peel / nb_samples
end

fraction_peel(0.1, 3)  # 0.26
fraction_peel(0.1, 10) # 0.65
fraction_peel(0.1, 15) # =-O stack overflow, too many rejections in `random_point_inside_unit_orange`