Isodiametric problem in Carnot groups
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1 Conference Geometric Measure Theory Université Paris Diderot, 12th-14th September 2012
2 Isodiametric inequality in R n Isodiametric inequality: where ω n = L n (B(0, 1)). L n (A) 2 n ω n (diam A) n
3 Isodiametric inequality in R n Isodiametric inequality: L n (A) 2 n ω n (diam A) n where ω n = L n (B(0, 1)). Equality holds iff A is a ball (up to a null set).
4 Isodiametric inequality in R n Isodiametric inequality: where ω n = L n (B(0, 1)). L n (A) 2 n ω n (diam A) n Equality holds iff A is a ball (up to a null set). An application: H n = S n = 2 n ωn 1 L n
5 Isodiametric inequality in R n where Isodiametric inequality: where ω n = L n (B(0, 1)). L n (A) 2 n ω n (diam A) n Equality holds iff A is a ball (up to a null set). An application: H n = S n = 2 n ωn 1 L n H n (A) = lim δ 0 inf{ i (diam A i ) n ; A i A i, diam A i δ} S n (A) = lim δ 0 inf{ i (diam A i ) n ; A i A i, A i ball, diam A i δ}
6 General setting Let G be a locally compact topological group equipped with:
7 General setting Let G be a locally compact topological group equipped with: dilations δ λ : G G, λ > 0, group homeomorphisms satisfying: δ 1 = Id δ λλ = δ λ δ λ
8 General setting Let G be a locally compact topological group equipped with: dilations δ λ : G G, λ > 0, group homeomorphisms satisfying: δ 1 = Id δ λλ = δ λ δ λ a left invariant homogeneous distance d inducing the topology of the group, i.e., satisfying: d(x.y, x.z) = d(y, z) d(δ λ (x), δ λ (y)) = λd(x, y)
9 Hausdorff measures Assume that for some Q 0 < H Q (B) < + for any ball B.
10 Hausdorff measures Assume that for some Q 0 < H Q (B) < + for any ball B. It follows that dim H (G) = Q, H Q and S Q are Haar measures on G and hence are proportional.
11 Hausdorff measures Assume that for some Q 0 < H Q (B) < + for any ball B. It follows that dim H (G) = Q, H Q and S Q are Haar measures on G and hence are proportional. Note that H Q (δ λ (A)) = λ Q H Q (A) and S Q (δ λ (A)) = λ Q S Q (A).
12 Hausdorff measures Proposition We have S Q (B) = (diam B) Q for any ball B.
13 Hausdorff measures Proposition We have S Q (B) = (diam B) Q for any ball B. Proposition We have S Q = C d H Q where { S Q } (A) C d = sup ; 0 < diam A < +. (diam A) Q
14 Isodiametric problem One seeks for the maximal possible value of the measure of sets with a given diameter:
15 Isodiametric problem One seeks for the maximal possible value of the measure of sets with a given diameter: where λ > 0 is fixed. sup{ S Q (A) ; diam A = λ }
16 Isodiametric problem One seeks for the maximal possible value of the measure of sets with a given diameter: where λ > 0 is fixed. Questions: Sup = Max? sup{ S Q (A) ; diam A = λ }
17 Isodiametric problem One seeks for the maximal possible value of the measure of sets with a given diameter: where λ > 0 is fixed. Questions: Sup = Max? Yes. sup{ S Q (A) ; diam A = λ }
18 Isodiametric problem One seeks for the maximal possible value of the measure of sets with a given diameter: sup{ S Q (A) ; diam A = λ } where λ > 0 is fixed. Questions: Sup = Max? Yes. Which are the sets, called isodiametric, that realize the max?
19 Isodiametric problem One seeks for the maximal possible value of the measure of sets with a given diameter: where λ > 0 is fixed. Questions: Sup = Max? Yes. sup{ S Q (A) ; diam A = λ } Which are the sets, called isodiametric, that realize the max? What kind of properties can be deduced from these informations?
20 Isodiametric problem One seeks for the maximal possible value of the measure of sets with a given diameter: where λ > 0 is fixed. Questions: Sup = Max? Yes. sup{ S Q (A) ; diam A = λ } Which are the sets, called isodiametric, that realize the max? What kind of properties can be deduced from these informations? Answers to these questions do not depend on the choice of a Haar measure.
21 Isodiametric problem One seeks for the maximal possible value of the measure of sets with a given diameter: where λ > 0 is fixed. Questions: Sup = Max? Yes. sup{ S Q (A) ; diam A = λ } Which are the sets, called isodiametric, that realize the max? What kind of properties can be deduced from these informations? Answers to these questions do not depend on the choice of a Haar measure. They may on the contrary depend strongly on the distance d the group G is equipped with.
22 Isodiametric problem Using dilations it holds: sup{s Q (A) ; diam A = λ} = λ Q sup{s Q (A) ; diam A = 1}
23 Isodiametric problem Using dilations it holds: and sup{s Q (A) ; diam A = λ} = λ Q sup{s Q (A) ; diam A = 1} sup{s Q (A); diam A = 1} { S Q } (A) = sup ; 0 < diam A < + = C (diam A) Q d.
24 Isodiametric problem Using dilations it holds: and sup{s Q (A) ; diam A = λ} = λ Q sup{s Q (A) ; diam A = 1} sup{s Q (A); diam A = 1} { S Q } (A) = sup ; 0 < diam A < + = C (diam A) Q d. Note that: C d 1,
25 Isodiametric problem Using dilations it holds: and sup{s Q (A) ; diam A = λ} = λ Q sup{s Q (A) ; diam A = 1} sup{s Q (A); diam A = 1} { S Q } (A) = sup ; 0 < diam A < + = C (diam A) Q d. Note that: C d 1, C d = 1 iff balls are isodiametric,
26 Isodiametric problem Using dilations it holds: and sup{s Q (A) ; diam A = λ} = λ Q sup{s Q (A) ; diam A = 1} sup{s Q (A); diam A = 1} { S Q } (A) = sup ; 0 < diam A < + = C (diam A) Q d. Note that: C d 1, C d = 1 iff balls are isodiametric, S Q = H Q iff balls are isodiametric.
27 Carnot groups Theorem [R] Let G be a non abelien Carnot group. There exists a left invariant homogeneous distance d on G such that C d > 1, i.e., for which balls are not isodiametric.
28 Example: Heisenberg group The Heisenberg group H n C n R
29 Example: Heisenberg group The Heisenberg group H n C n R group law: [z, t] [z, t ] = [z + z, t + t + 2 Im zz ] dilations: δ λ ([z, t]) = [λz, λ 2 t]
30 Example: Heisenberg group The Heisenberg group H n C n R group law: [z, t] [z, t ] = [z + z, t + t + 2 Im zz ] dilations: δ λ ([z, t]) = [λz, λ 2 t] with the left invariant homogeneous distance d ([z, t], [z, t ]) = [z, t] 1 [z, t ] where [z, t] = max( z, t 1/2 ).
31 Carnot groups A Carnot group G is a connected and simply connected nilpotent Lie group whose Lie algebra G admits a stratification, G = k j=1v j, [V 1, V j ] = V j+1, V k {0}, V k+1 = {0}, for some integer k 1 called the step of the stratification.
32 Carnot groups A Carnot group G is a connected and simply connected nilpotent Lie group whose Lie algebra G admits a stratification, G = k j=1v j, [V 1, V j ] = V j+1, V k {0}, V k+1 = {0}, for some integer k 1 called the step of the stratification. The exponential map exp : G G is a global diffeomorphism and the group law is given by the Campbell-Hausdorff formula, exp X exp Y = exp H(X, Y ), where H(X, Y ) = X + Y + [X, Y ]/2 +.
33 Carnot groups A Carnot group G is a connected and simply connected nilpotent Lie group whose Lie algebra G admits a stratification, G = k j=1v j, [V 1, V j ] = V j+1, V k {0}, V k+1 = {0}, for some integer k 1 called the step of the stratification. The exponential map exp : G G is a global diffeomorphism and the group law is given by the Campbell-Hausdorff formula, exp X exp Y = exp H(X, Y ), where H(X, Y ) = X + Y + [X, Y ]/2 +. Dilations on G are given by δ λ ( k j=1 Y j) = k j=1 λj Y j, Y j V j, λ > 0.
34 Carnot groups A Carnot group G is a connected and simply connected nilpotent Lie group whose Lie algebra G admits a stratification, G = k j=1v j, [V 1, V j ] = V j+1, V k {0}, V k+1 = {0}, for some integer k 1 called the step of the stratification. The exponential map exp : G G is a global diffeomorphism and the group law is given by the Campbell-Hausdorff formula, exp X exp Y = exp H(X, Y ), where H(X, Y ) = X + Y + [X, Y ]/2 +. Dilations on G are given by δ λ ( k j=1 Y j) = k j=1 λj Y j, Y j V j, λ > 0. Homogeneous dimension: Q = k j=1 j dim V j.
35 A left invariant homogeneous distance d Let (X 1,..., X n ) be a basis of G adapted to the stratification and define an Euclidean norm on G by declaring it orthonormal.
36 A left invariant homogeneous distance d Let (X 1,..., X n ) be a basis of G adapted to the stratification and define an Euclidean norm on G by declaring it orthonormal. Choose positive coefficients c j so that H(Y, Z) Y + Z where Y = max j c j Y j 1/j whenever Y = Y Y k, Y j V j.
37 A left invariant homogeneous distance d Let (X 1,..., X n ) be a basis of G adapted to the stratification and define an Euclidean norm on G by declaring it orthonormal. Choose positive coefficients c j so that H(Y, Z) Y + Z where Y = max j c j Y j 1/j whenever Y = Y Y k, Y j V j. Set x = exp 1 x and d (x, y) = x 1 y.
38 A left invariant homogeneous distance d Let (X 1,..., X n ) be a basis of G adapted to the stratification and define an Euclidean norm on G by declaring it orthonormal. Choose positive coefficients c j so that H(Y, Z) Y + Z where Y = max j c j Y j 1/j whenever Y = Y Y k, Y j V j. Set x = exp 1 x and d (x, y) = x 1 y. Then d is a left invariant homogeneous distance on G.
39 A left invariant homogeneous distance d Let (X 1,..., X n ) be a basis of G adapted to the stratification and define an Euclidean norm on G by declaring it orthonormal. Choose positive coefficients c j so that H(Y, Z) Y + Z where Y = max j c j Y j 1/j whenever Y = Y Y k, Y j V j. Set x = exp 1 x and d (x, y) = x 1 y. Then d is a left invariant homogeneous distance on G. Examples: The abelian case (k = 1): (R n,+) equipped with the usual dialations δ λ (x) = λx and the Euclidean distance.
40 A left invariant homogeneous distance d Let (X 1,..., X n ) be a basis of G adapted to the stratification and define an Euclidean norm on G by declaring it orthonormal. Choose positive coefficients c j so that H(Y, Z) Y + Z where Y = max j c j Y j 1/j whenever Y = Y Y k, Y j V j. Set x = exp 1 x and d (x, y) = x 1 y. Then d is a left invariant homogeneous distance on G. Examples: The abelian case (k = 1): (R n,+) equipped with the usual dialations δ λ (x) = λx and the Euclidean distance. In H n : [z, t] = max( z, t 1/2 ).
41 Carnot groups Theorem [R] Let G be a non abelien Carnot group (k 2). Then C d > 1, i.e., balls in (G, d ) are not isodiametric.
42 Carnot groups Theorem [R] Let G be a non abelien Carnot group (k 2). Then C d > 1, i.e., balls in (G, d ) are not isodiametric. Proof. A sufficient condition for not being an isodiametric set in (G, d): Assume that A is compact and diam A > 0. Assume there exists x A such that d(x, y) < diam A for all y A. Then A is not isodiametric.
43 Carnot groups Theorem [R] Let G be a non abelien Carnot group (k 2). Then C d > 1, i.e., balls in (G, d ) are not isodiametric. Proof. A sufficient condition for not being an isodiametric set in (G, d): Assume that A is compact and diam A > 0. Assume there exists x A such that d(x, y) < diam A for all y A. Then A is not isodiametric. Take d = d and B a ball centered at 0. Apply the lemma to x = exp X B with X V k.
44 Carnot groups Corollary Let G be a non abelien Carnot group equipped with some homogeneous distance. Let Q = dim H (G). Then G is purely Q-unrectifiable.
45 Carnot groups Corollary Let G be a non abelien Carnot group equipped with some homogeneous distance. Let Q = dim H (G). Then G is purely Q-unrectifiable. Otherwise one can find a Lipschitz map f : A R Q (G, d ) such that 0 < H Q (f (A)) < +. Then it holds H Q (f (A) B(x, r)) lim r 0 (2r) Q = 1 for H Q a.e. x f (A).
46 Carnot groups Corollary Let G be a non abelien Carnot group equipped with some homogeneous distance. Let Q = dim H (G). Then G is purely Q-unrectifiable. Otherwise one can find a Lipschitz map f : A R Q (G, d ) such that 0 < H Q (f (A)) < +. Then it holds H Q (f (A) B(x, r)) lim r 0 (2r) Q = 1 for H Q a.e. x f (A). On the other hand, H Q (f (A) B(x, r)) (2r) Q HQ (B(x, r)) (2r) Q = HQ (B(x, r)) S Q (B(x, r)) = C 1 d < 1, which gives a contradiction.
47 CC-distances Fix a left invariant Riemannian metric g on G and set d c (x, y) = inf{length g (γ); γ horizontal curve joining x to y}, where γ is said to be horizontal if it is absolutely continuous and such that γ(t) span{x (γ(t)); X V 1 } a.e.
48 CC-distances Fix a left invariant Riemannian metric g on G and set d c (x, y) = inf{length g (γ); γ horizontal curve joining x to y}, where γ is said to be horizontal if it is absolutely continuous and such that γ(t) span{x (γ(t)); X V 1 } a.e. d c is a left invariant homogeneous distance on G called Carnot-Carathéodory distance.
49 CC-distances Fix a left invariant Riemannian metric g on G and set d c (x, y) = inf{length g (γ); γ horizontal curve joining x to y}, where γ is said to be horizontal if it is absolutely continuous and such that γ(t) span{x (γ(t)); X V 1 } a.e. d c is a left invariant homogeneous distance on G called Carnot-Carathéodory distance. Example: H n with a Riemannian metric making (X 1,..., X n, Y 1,..., Y n ) an othonormal basis of the first layer where X j = xj + 2x n+j t, Y j = xn+j 2x j t (with z j = x j + ix j+n ).
50 Carnot groups equipped with CC-distance Theorem [R] Let G be a non abelien Carnot group equipped with a Carnot-Carathéodory distance d c. Assume that there exists a length minimizing curve γ : [a, b] G that stops to be minimizing after reaching γ(b). Then balls are not isodiametric.
51 Carnot groups equipped with CC-distance Theorem [R] Let G be a non abelien Carnot group equipped with a Carnot-Carathéodory distance d c. Assume that there exists a length minimizing curve γ : [a, b] G that stops to be minimizing after reaching γ(b). Then balls are not isodiametric. Example: H n
52 Carnot groups equipped with CC-distance Theorem [R] Let G be a non abelien Carnot group equipped with a Carnot-Carathéodory distance d c. Assume that there exists a length minimizing curve γ : [a, b] G that stops to be minimizing after reaching γ(b). Then balls are not isodiametric. Example: H n Does there exist a Carnot group G and a homogeneous distance d on G such that C d = 1?
53 Besicovitch 1/2-problem Given a metric space (M, d), let σ n (M, d) denote the smallest number such that every subset A M of finite H n -measure having at H n -a.e. x A is n-rectifiable, where D n (A, x) > σ n (M, d) D n (A, x) = lim inf r 0 H n (A B(x, r)) (2r) n.
54 Besicovitch 1/2-problem Given a metric space (M, d), let σ n (M, d) denote the smallest number such that every subset A M of finite H n -measure having at H n -a.e. x A is n-rectifiable, where D n (A, x) > σ n (M, d) D n (A, x) = lim inf r 0 H n (A B(x, r)) (2r) n. NB: σ n (M, d) 1 and σ n (M, d) = 1 does not give any significant information about rectifiability.
55 Besicovitch 1/2-problem Given a metric space (M, d), let σ n (M, d) denote the smallest number such that every subset A M of finite H n -measure having at H n -a.e. x A is n-rectifiable, where D n (A, x) > σ n (M, d) D n (A, x) = lim inf r 0 H n (A B(x, r)) (2r) n. NB: σ n (M, d) 1 and σ n (M, d) = 1 does not give any significant information about rectifiability. Besicovitch 1/2-problem: σ n (M, d) 1/2?
56 Besicovitch 1/2-problem Corollary Let (G, d) be a non abelian Carnot group equipped with a homogeneous distance. Let Q = dim H (G). Then one has σ Q (G, d) = C 1 d. In particular σ Q (G, d) < 1 iff balls are not isodiametric.
57 Besicovitch 1/2-problem in H n Theorem In (H n, d ), we have 1 < C d < 2 and hence 1/2 < σ 2n+2 (H n, d ) < 1. In (H n, d c ), we have C dc > 1 and hence σ 2n+2 (H n, d c ) < 1. If n 8, then C dc < 2 and hence σ 2n+2 (H n, d c ) > 1/2.
58 Isodiametric sets in (H n, d c ) (joint work with G.P. Leonardi and D. Vittone) Unit ball in (H n, d c )
59 Isodiametric sets in (H n, d c ) (joint work with G.P. Leonardi and D. Vittone) The set A 2
60 Rotationally invariant isodiametric sets in (H n, d c ) (joint work with G.P. Leonardi and D. Vittone) Given θ = (θ 1,..., θ n ) R n, let r θ ([z, t]) = [(e iθ 1 z 1,..., e iθn z n ), t]
61 Rotationally invariant isodiametric sets in (H n, d c ) (joint work with G.P. Leonardi and D. Vittone) Given θ = (θ 1,..., θ n ) R n, let r θ ([z, t]) = [(e iθ 1 z 1,..., e iθn z n ), t] R = {F H n ; r θ (F ) F for all θ R n }
62 Rotationally invariant isodiametric sets in (H n, d c ) (joint work with G.P. Leonardi and D. Vittone) Given θ = (θ 1,..., θ n ) R n, let r θ ([z, t]) = [(e iθ 1 z 1,..., e iθn z n ), t] R = {F H n ; r θ (F ) F for all θ R n } { S 2n+2 } (F ) C dc,r = sup ; F R, 0 < diam F < + (diam F ) 2n+2
63 Rotationally invariant isodiametric sets in (H n, d c ) (joint work with G.P. Leonardi and D. Vittone) Given θ = (θ 1,..., θ n ) R n, let r θ ([z, t]) = [(e iθ 1 z 1,..., e iθn z n ), t] R = {F H n ; r θ (F ) F for all θ R n } { S 2n+2 } (F ) C dc,r = sup ; F R, 0 < diam F < + (diam F ) 2n+2 I R = {F R ; F compact, diam F > 0, S 2n+2 (F ) = C dc,r (diam F ) 2n+2 }.
64 Rotationally invariant isodiametric sets in (H n, d c ) (joint work with G.P. Leonardi and D. Vittone) Given θ = (θ 1,..., θ n ) R n, let r θ ([z, t]) = [(e iθ 1 z 1,..., e iθn z n ), t] R = {F H n ; r θ (F ) F for all θ R n } { S 2n+2 } (F ) C dc,r = sup ; F R, 0 < diam F < + (diam F ) 2n+2 I R = {F R ; F compact, diam F > 0, S 2n+2 (F ) = C dc,r (diam F ) 2n+2 }. For F H n, let St F denote its Steiner symetrisation w.r.t. the C n -plane.
65 Rotationally invariant isodiametric sets in (H n, d c ) (joint work with G.P. Leonardi and D. Vittone) Theorem [Leonardi-R-Vittone] Let E I R then St E I R and St E coincides with the set A diam E.
66 Rotationally invariant isodiametric sets in (H n, d c ) (joint work with G.P. Leonardi and D. Vittone) Theorem [Leonardi-R-Vittone] Let E I R then St E I R and St E coincides with the set A diam E. Corollary (non uniqueness in I R ) There exists E I R such that p E does not coincide with A diam E for all p H n.
67 Rotationally invariant isodiametric sets in (H n, d c ) (joint work with G.P. Leonardi and D. Vittone) Theorem [Leonardi-R-Vittone] Let E I R then St E I R and St E coincides with the set A diam E. Corollary (non uniqueness in I R ) There exists E I R such that p E does not coincide with A diam E for all p H n. Corollary (existence of non rotationally invariant isodiametric sets) There exists an isodiametric set E that does not belong to R.
68 Small perturbation of A A small pertubation of A 2 with same diameter and same volume.
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