Theory HOL-Analysis.Ball_Volume
section ‹The Volume of an ‹n›-Dimensional Ball›
theory Ball_Volume
imports Gamma_Function Lebesgue_Integral_Substitution
begin
text ‹
We define the volume of the unit ball in terms of the Gamma function. Note that the
dimension need not be an integer; we also allow fractional dimensions, although we do
not use this case or prove anything about it for now.
›
definition unit_ball_vol :: "real ⇒ real" where
"unit_ball_vol n = pi powr (n / 2) / Gamma (n / 2 + 1)"
lemma unit_ball_vol_pos [simp]: "n ≥ 0 ⟹ unit_ball_vol n > 0"
by (force simp: unit_ball_vol_def intro: divide_nonneg_pos)
lemma unit_ball_vol_nonneg [simp]: "n ≥ 0 ⟹ unit_ball_vol n ≥ 0"
by (simp add: dual_order.strict_implies_order)
text ‹
We first need the value of the following integral, which is at the core of
computing the measure of an ‹n + 1›-dimensional ball in terms of the measure of an
‹n›-dimensional one.
›
lemma emeasure_cball_aux_integral:
"(∫⇧+x. indicator {-1..1} x * sqrt (1 - x⇧2) ^ n ∂lborel) =
ennreal (Beta (1 / 2) (real n / 2 + 1))"
proof -
have "((λt. t powr (-1 / 2) * (1 - t) powr (real n / 2)) has_integral
Beta (1 / 2) (real n / 2 + 1)) {0..1}"
using has_integral_Beta_real[of "1/2" "n / 2 + 1"] by simp
from nn_integral_has_integral_lebesgue[OF _ this] have
"ennreal (Beta (1 / 2) (real n / 2 + 1)) =
nn_integral lborel (λt. ennreal (t powr (-1 / 2) * (1 - t) powr (real n / 2) *
indicator {0^2..1^2} t))"
by (simp add: mult_ac ennreal_mult' ennreal_indicator)
also have "… = (∫⇧+ x. ennreal (x⇧2 powr - (1 / 2) * (1 - x⇧2) powr (real n / 2) * (2 * x) *
indicator {0..1} x) ∂lborel)"
by (subst nn_integral_substitution[where g = "λx. x ^ 2" and g' = "λx. 2 * x"])
(auto intro!: derivative_eq_intros continuous_intros simp: set_borel_measurable_def)
also have "… = (∫⇧+ x. 2 * ennreal ((1 - x⇧2) powr (real n / 2) * indicator {0..1} x) ∂lborel)"
by (intro nn_integral_cong_AE AE_I[of _ _ "{0}"])
(auto simp: indicator_def powr_minus powr_half_sqrt field_split_simps ennreal_mult')
also have "… = (∫⇧+ x. ennreal ((1 - x⇧2) powr (real n / 2) * indicator {0..1} x) ∂lborel) +
(∫⇧+ x. ennreal ((1 - x⇧2) powr (real n / 2) * indicator {0..1} x) ∂lborel)"
(is "_ = ?I + _") by (simp add: mult_2 nn_integral_add)
also have "?I = (∫⇧+ x. ennreal ((1 - x⇧2) powr (real n / 2) * indicator {-1..0} x) ∂lborel)"
by (subst nn_integral_real_affine[of _ "-1" 0])
(auto simp: indicator_def intro!: nn_integral_cong)
hence "?I + ?I = … + ?I" by simp
also have "… = (∫⇧+ x. ennreal ((1 - x⇧2) powr (real n / 2) *
(indicator {-1..0} x + indicator{0..1} x)) ∂lborel)"
by (subst nn_integral_add [symmetric]) (auto simp: algebra_simps)
also have "… = (∫⇧+ x. ennreal ((1 - x⇧2) powr (real n / 2) * indicator {-1..1} x) ∂lborel)"
by (intro nn_integral_cong_AE AE_I[of _ _ "{0}"]) (auto simp: indicator_def)
also have "… = (∫⇧+ x. ennreal (indicator {-1..1} x * sqrt (1 - x⇧2) ^ n) ∂lborel)"
by (intro nn_integral_cong_AE AE_I[of _ _ "{1, -1}"])
(auto simp: powr_half_sqrt [symmetric] indicator_def abs_square_le_1
abs_square_eq_1 powr_def exp_of_nat_mult [symmetric] emeasure_lborel_countable)
finally show ?thesis ..
qed
lemma real_sqrt_le_iff': "x ≥ 0 ⟹ y ≥ 0 ⟹ sqrt x ≤ y ⟷ x ≤ y ^ 2"
using real_le_lsqrt sqrt_le_D by blast
text ‹
Isabelle's type system makes it very difficult to do an induction over the dimension
of a Euclidean space type, because the type would change in the inductive step. To avoid
this problem, we instead formulate the problem in a more concrete way by unfolding the
definition of the Euclidean norm.
›
lemma emeasure_cball_aux:
assumes "finite A" "r > 0"
shows "emeasure (Pi⇩M A (λ_. lborel))
({f. sqrt (∑i∈A. (f i)⇧2) ≤ r} ∩ space (Pi⇩M A (λ_. lborel))) =
ennreal (unit_ball_vol (real (card A)) * r ^ card A)"
using assms
proof (induction arbitrary: r)
case (empty r)
thus ?case
by (simp add: unit_ball_vol_def space_PiM)
next
case (insert i A r)
interpret product_sigma_finite "λ_. lborel"
by standard
have "emeasure (Pi⇩M (insert i A) (λ_. lborel))
({f. sqrt (∑i∈insert i A. (f i)⇧2) ≤ r} ∩ space (Pi⇩M (insert i A) (λ_. lborel))) =
nn_integral (Pi⇩M (insert i A) (λ_. lborel))
(indicator ({f. sqrt (∑i∈insert i A. (f i)⇧2) ≤ r} ∩
space (Pi⇩M (insert i A) (λ_. lborel))))"
by (subst nn_integral_indicator) auto
also have "… = (∫⇧+ y. ∫⇧+ x. indicator ({f. sqrt ((f i)⇧2 + (∑i∈A. (f i)⇧2)) ≤ r} ∩
space (Pi⇩M (insert i A) (λ_. lborel))) (x(i := y))
∂Pi⇩M A (λ_. lborel) ∂lborel)"
using insert.prems insert.hyps by (subst product_nn_integral_insert_rev) auto
also have "… = (∫⇧+ (y::real). ∫⇧+ x. indicator {-r..r} y * indicator ({f. sqrt ((∑i∈A. (f i)⇧2)) ≤
sqrt (r ^ 2 - y ^ 2)} ∩ space (Pi⇩M A (λ_. lborel))) x ∂Pi⇩M A (λ_. lborel) ∂lborel)"
proof (intro nn_integral_cong, goal_cases)
case (1 y f)
have *: "y ∈ {-r..r}" if "y ^ 2 + c ≤ r ^ 2" "c ≥ 0" for c
proof -
have "y ^ 2 ≤ y ^ 2 + c" using that by simp
also have "… ≤ r ^ 2" by fact
finally show ?thesis
using ‹r > 0› by (simp add: power2_le_iff_abs_le abs_if split: if_splits)
qed
have "(∑x∈A. (if x = i then y else f x)⇧2) = (∑x∈A. (f x)⇧2)"
using insert.hyps by (intro sum.cong) auto
thus ?case using 1 ‹r > 0›
by (auto simp: sum_nonneg real_sqrt_le_iff' indicator_def PiE_def space_PiM dest!: *)
qed
also have "… = (∫⇧+ (y::real). indicator {-r..r} y * (∫⇧+ x. indicator ({f. sqrt ((∑i∈A. (f i)⇧2))
≤ sqrt (r ^ 2 - y ^ 2)} ∩ space (Pi⇩M A (λ_. lborel))) x
∂Pi⇩M A (λ_. lborel)) ∂lborel)" by (subst nn_integral_cmult) auto
also have "… = (∫⇧+ (y::real). indicator {-r..r} y * emeasure (PiM A (λ_. lborel))
({f. sqrt ((∑i∈A. (f i)⇧2)) ≤ sqrt (r ^ 2 - y ^ 2)} ∩ space (Pi⇩M A (λ_. lborel))) ∂lborel)"
using ‹finite A› by (intro nn_integral_cong, subst nn_integral_indicator) auto
also have "… = (∫⇧+ (y::real). indicator {-r..r} y * ennreal (unit_ball_vol (real (card A)) *
(sqrt (r ^ 2 - y ^ 2)) ^ card A) ∂lborel)"
proof (intro nn_integral_cong_AE, goal_cases)
case 1
have "AE y in lborel. y ∉ {-r,r}"
by (intro AE_not_in countable_imp_null_set_lborel) auto
thus ?case
proof eventually_elim
case (elim y)
show ?case
proof (cases "y ∈ {-r<..<r}")
case True
hence "y⇧2 < r⇧2" by (subst real_sqrt_less_iff [symmetric]) auto
thus ?thesis by (subst insert.IH) (auto)
qed (insert elim, auto)
qed
qed
also have "… = ennreal (unit_ball_vol (real (card A))) *
(∫⇧+ (y::real). indicator {-r..r} y * (sqrt (r ^ 2 - y ^ 2)) ^ card A ∂lborel)"
by (subst nn_integral_cmult [symmetric])
(auto simp: mult_ac ennreal_mult' [symmetric] indicator_def intro!: nn_integral_cong)
also have "(∫⇧+ (y::real). indicator {-r..r} y * (sqrt (r ^ 2 - y ^ 2)) ^ card A ∂lborel) =
(∫⇧+ (y::real). r ^ card A * indicator {-1..1} y * (sqrt (1 - y ^ 2)) ^ card A
∂(distr lborel borel ((*) (1/r))))" using ‹r > 0›
by (subst nn_integral_distr)
(auto simp: indicator_def field_simps real_sqrt_divide intro!: nn_integral_cong)
also have "… = (∫⇧+ x. ennreal (r ^ Suc (card A)) *
(indicator {- 1..1} x * sqrt (1 - x⇧2) ^ card A) ∂lborel)" using ‹r > 0›
by (subst lborel_distr_mult) (auto simp: nn_integral_density ennreal_mult' [symmetric] mult_ac)
also have "… = ennreal (r ^ Suc (card A)) * (∫⇧+ x. indicator {- 1..1} x *
sqrt (1 - x⇧2) ^ card A ∂lborel)"
by (subst nn_integral_cmult) auto
also note emeasure_cball_aux_integral
also have "ennreal (unit_ball_vol (real (card A))) * (ennreal (r ^ Suc (card A)) *
ennreal (Beta (1/2) (card A / 2 + 1))) =
ennreal (unit_ball_vol (card A) * Beta (1/2) (card A / 2 + 1) * r ^ Suc (card A))"
using ‹r > 0› by (simp add: ennreal_mult' [symmetric] mult_ac)
also have "unit_ball_vol (card A) * Beta (1/2) (card A / 2 + 1) = unit_ball_vol (Suc (card A))"
by (auto simp: unit_ball_vol_def Beta_def Gamma_eq_zero_iff field_simps
Gamma_one_half_real powr_half_sqrt [symmetric] powr_add [symmetric])
also have "Suc (card A) = card (insert i A)" using insert.hyps by simp
finally show ?case .
qed
text ‹
We now get the main theorem very easily by just applying the above lemma.
›
context
fixes c :: "'a :: euclidean_space" and r :: real
assumes r: "r ≥ 0"
begin
theorem emeasure_cball:
"emeasure lborel (cball c r) = ennreal (unit_ball_vol (DIM('a)) * r ^ DIM('a))"
proof (cases "r = 0")
case False
with r have r: "r > 0" by simp
have "(lborel :: 'a measure) =
distr (Pi⇩M Basis (λ_. lborel)) borel (λf. ∑b∈Basis. f b *⇩R b)"
by (rule lborel_eq)
also have "emeasure … (cball 0 r) =
emeasure (Pi⇩M Basis (λ_. lborel))
({y. dist 0 (∑b∈Basis. y b *⇩R b :: 'a) ≤ r} ∩ space (Pi⇩M Basis (λ_. lborel)))"
by (subst emeasure_distr) (auto simp: cball_def)
also have "{f. dist 0 (∑b∈Basis. f b *⇩R b :: 'a) ≤ r} = {f. sqrt (∑i∈Basis. (f i)⇧2) ≤ r}"
by (subst euclidean_dist_l2) (auto simp: L2_set_def)
also have "emeasure (Pi⇩M Basis (λ_. lborel)) (… ∩ space (Pi⇩M Basis (λ_. lborel))) =
ennreal (unit_ball_vol (real DIM('a)) * r ^ DIM('a))"
using r by (subst emeasure_cball_aux) simp_all
also have "emeasure lborel (cball 0 r :: 'a set) =
emeasure (distr lborel borel (λx. c + x)) (cball c r)"
by (subst emeasure_distr) (auto simp: cball_def dist_norm norm_minus_commute)
also have "distr lborel borel (λx. c + x) = lborel"
using lborel_affine[of 1 c] by (simp add: density_1)
finally show ?thesis .
qed auto
corollary content_cball:
"content (cball c r) = unit_ball_vol (DIM('a)) * r ^ DIM('a)"
by (simp add: measure_def emeasure_cball r)
corollary emeasure_ball:
"emeasure lborel (ball c r) = ennreal (unit_ball_vol (DIM('a)) * r ^ DIM('a))"
proof -
from negligible_sphere[of c r] have "sphere c r ∈ null_sets lborel"
by (auto simp: null_sets_completion_iff negligible_iff_null_sets negligible_convex_frontier)
hence "emeasure lborel (ball c r ∪ sphere c r :: 'a set) = emeasure lborel (ball c r :: 'a set)"
by (intro emeasure_Un_null_set) auto
also have "ball c r ∪ sphere c r = (cball c r :: 'a set)" by auto
also have "emeasure lborel … = ennreal (unit_ball_vol (real DIM('a)) * r ^ DIM('a))"
by (rule emeasure_cball)
finally show ?thesis ..
qed
corollary content_ball:
"content (ball c r) = unit_ball_vol (DIM('a)) * r ^ DIM('a)"
by (simp add: measure_def r emeasure_ball)
end
text ‹
Lastly, we now prove some nicer explicit formulas for the volume of the unit balls in
the cases of even and odd integer dimensions.
›
lemma unit_ball_vol_even:
"unit_ball_vol (real (2 * n)) = pi ^ n / fact n"
by (simp add: unit_ball_vol_def add_ac powr_realpow Gamma_fact)
lemma unit_ball_vol_odd':
"unit_ball_vol (real (2 * n + 1)) = pi ^ n / pochhammer (1 / 2) (Suc n)"
and unit_ball_vol_odd:
"unit_ball_vol (real (2 * n + 1)) =
(2 ^ (2 * Suc n) * fact (Suc n)) / fact (2 * Suc n) * pi ^ n"
proof -
have "unit_ball_vol (real (2 * n + 1)) =
pi powr (real n + 1 / 2) / Gamma (1 / 2 + real (Suc n))"
by (simp add: unit_ball_vol_def field_simps)
also have "pochhammer (1 / 2) (Suc n) = Gamma (1 / 2 + real (Suc n)) / Gamma (1 / 2)"
by (intro pochhammer_Gamma) auto
hence "Gamma (1 / 2 + real (Suc n)) = sqrt pi * pochhammer (1 / 2) (Suc n)"
by (simp add: Gamma_one_half_real)
also have "pi powr (real n + 1 / 2) / … = pi ^ n / pochhammer (1 / 2) (Suc n)"
by (simp add: powr_add powr_half_sqrt powr_realpow)
finally show "unit_ball_vol (real (2 * n + 1)) = …" .
also have "pochhammer (1 / 2 :: real) (Suc n) =
fact (2 * Suc n) / (2 ^ (2 * Suc n) * fact (Suc n))"
using fact_double[of "Suc n", where ?'a = real] by (simp add: divide_simps mult_ac)
also have "pi ^n / … = (2 ^ (2 * Suc n) * fact (Suc n)) / fact (2 * Suc n) * pi ^ n"
by simp
finally show "unit_ball_vol (real (2 * n + 1)) = …" .
qed
lemma unit_ball_vol_numeral:
"unit_ball_vol (numeral (Num.Bit0 n)) = pi ^ numeral n / fact (numeral n)" (is ?th1)
"unit_ball_vol (numeral (Num.Bit1 n)) = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) /
fact (2 * Suc (numeral n)) * pi ^ numeral n" (is ?th2)
proof -
have "numeral (Num.Bit0 n) = (2 * numeral n :: nat)"
by (simp only: numeral_Bit0 mult_2 ring_distribs)
also have "unit_ball_vol … = pi ^ numeral n / fact (numeral n)"
by (rule unit_ball_vol_even)
finally show ?th1 by simp
next
have "numeral (Num.Bit1 n) = (2 * numeral n + 1 :: nat)"
by (simp only: numeral_Bit1 mult_2)
also have "unit_ball_vol … = 2 ^ (2 * Suc (numeral n)) * fact (Suc (numeral n)) /
fact (2 * Suc (numeral n)) * pi ^ numeral n"
by (rule unit_ball_vol_odd)
finally show ?th2 by simp
qed
lemmas eval_unit_ball_vol = unit_ball_vol_numeral fact_numeral
text ‹
Just for fun, we compute the volume of unit balls for a few dimensions.
›
lemma unit_ball_vol_0 [simp]: "unit_ball_vol 0 = 1"
using unit_ball_vol_even[of 0] by simp
lemma unit_ball_vol_1 [simp]: "unit_ball_vol 1 = 2"
using unit_ball_vol_odd[of 0] by simp
corollary
unit_ball_vol_2: "unit_ball_vol 2 = pi"
and unit_ball_vol_3: "unit_ball_vol 3 = 4 / 3 * pi"
and unit_ball_vol_4: "unit_ball_vol 4 = pi⇧2 / 2"
and unit_ball_vol_5: "unit_ball_vol 5 = 8 / 15 * pi⇧2"
by (simp_all add: eval_unit_ball_vol)
corollary circle_area:
"r ≥ 0 ⟹ content (ball c r :: (real ^ 2) set) = r ^ 2 * pi"
by (simp add: content_ball unit_ball_vol_2)
corollary sphere_volume:
"r ≥ 0 ⟹ content (ball c r :: (real ^ 3) set) = 4 / 3 * r ^ 3 * pi"
by (simp add: content_ball unit_ball_vol_3)
text ‹
Useful equivalent forms
›
corollary content_ball_eq_0_iff [simp]: "content (ball c r) = 0 ⟷ r ≤ 0"
proof -
have "r > 0 ⟹ content (ball c r) > 0"
by (simp add: content_ball unit_ball_vol_def)
then show ?thesis
by (fastforce simp: ball_empty)
qed
corollary content_ball_gt_0_iff [simp]: "0 < content (ball z r) ⟷ 0 < r"
by (auto simp: zero_less_measure_iff)
corollary content_cball_eq_0_iff [simp]: "content (cball c r) = 0 ⟷ r ≤ 0"
proof (cases "r = 0")
case False
moreover have "r > 0 ⟹ content (cball c r) > 0"
by (simp add: content_cball unit_ball_vol_def)
ultimately show ?thesis
by fastforce
qed auto
corollary content_cball_gt_0_iff [simp]: "0 < content (cball z r) ⟷ 0 < r"
by (auto simp: zero_less_measure_iff)
end