HYPERSPACES OF EUCLIDEAN SPACES IN THE GROMOV-HAUSDORFF METRIC - - PowerPoint PPT Presentation

HYPERSPACES OF EUCLIDEAN SPACES IN THE GROMOV-HAUSDORFF METRIC

SERGEY A. ANTONYAN

National University of Mexico

12th Symposium on General Topology and its Relations to Modern Analysis and Algebra July 25-29, 2016 Prague Czech Republic

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 1 / 29

1

The Gromov-Hausdorff distance

2

The Urysohn space

3

The Euclidean-Hausdorff distance

4

Main Results

5

The Chebyshev balls

6

Orbit spaces of Hyperspaces

7

Properties of Ch(n)

8

Some ideas of the proof

9

Equivariant DDP

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 2 / 29

The Gromov-Hausdorff distance

Definition

Let (M, d) be a metric space. For two subsets A, B ⊂ M, the Hausdorff distance dH(A, B) is defined as follows: dH(A, B) = max{sup

a∈A

d(a, B), sup

b∈B

d(b, A)}.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 3 / 29

The Gromov-Hausdorff distance

Definition

Let (M, d) be a metric space. For two subsets A, B ⊂ M, the Hausdorff distance dH(A, B) is defined as follows: dH(A, B) = max{sup

a∈A

d(a, B), sup

b∈B

d(b, A)}. 2M denotes the set of all nonempty compact subsets of M. (2M, dH) is a metric space.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 3 / 29

The Gromov-Hausdorff distance dGH is a useful tool for studying topological properties of families of metric spaces. M. Gromov first introduced the notion of Gromov-Hausdorff distance in his ICM 1979 address in Helsinki on synthetic Riemannian geometry. Two years later dGH appeared in the book M.Gromov [3]. It turns the set GH of all isometry classes of compact metric spaces into a metric space. For two compact metric spaces X and Y the number dGH(X, Y) is defined to be the infimum of all Hausdorff distances dH(i(X), j(Y)) for all metric spaces M and all isometric embeddings i : X ֒ → M and j : Y ֒ → M. dGH(X, Y) = inf{dH(i(X), j(Y)) | i : X ֒ → M, j : Y ֒ → M}.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 4 / 29

Clearly, the Gromov-Hausdorff distance between isometric spaces is zero; it is a metric on the family GH of isometry classes of compact metric spaces. The metric “space” (GH, dGH) is called the Gromov-Hausdorff space.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 5 / 29

Clearly, the Gromov-Hausdorff distance between isometric spaces is zero; it is a metric on the family GH of isometry classes of compact metric spaces. The metric “space” (GH, dGH) is called the Gromov-Hausdorff space.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 5 / 29

Urysohn universal metric space

Theorem (Urysohn, 1925)

There exists, up to isometry, unique metric space U satisfying the following properties:

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 6 / 29

Urysohn universal metric space

Theorem (Urysohn, 1925)

There exists, up to isometry, unique metric space U satisfying the following properties:

1

U is Polish, i.e., separable and complete,

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 6 / 29

Urysohn universal metric space

Theorem (Urysohn, 1925)

There exists, up to isometry, unique metric space U satisfying the following properties:

1

U is Polish, i.e., separable and complete,

2

U contains an isometric copy of every separable metric space,

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 6 / 29

Urysohn universal metric space

Theorem (Urysohn, 1925)

There exists, up to isometry, unique metric space U satisfying the following properties:

1

U is Polish, i.e., separable and complete,

2

U contains an isometric copy of every separable metric space,

3

U is ultrahomogeneous, i.e., any isometry f : A → B between two finite subspaces of U, extends to an isometry F : U → U.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 6 / 29

Urysohn universal metric space

Theorem (Urysohn, 1925)

There exists, up to isometry, unique metric space U satisfying the following properties:

1

U is Polish, i.e., separable and complete,

2

U contains an isometric copy of every separable metric space,

3

U is ultrahomogeneous, i.e., any isometry f : A → B between two finite subspaces of U, extends to an isometry F : U → U. U is called the Urysohn universal metric space.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 6 / 29

Urysohn universal metric space

Theorem (Urysohn, 1925)

There exists, up to isometry, unique metric space U satisfying the following properties:

1

U is Polish, i.e., separable and complete,

2

U contains an isometric copy of every separable metric space,

3

U is ultrahomogeneous, i.e., any isometry f : A → B between two finite subspaces of U, extends to an isometry F : U → U. U is called the Urysohn universal metric space.

Theorem (Huhunaishvili, 1955)

The property (3) holds true for compact isometric subsets A ⊂ U, B⊂U.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 6 / 29

The Urysohn space U was introdused by Pavel S. Urysohn shortly before his tragic death in 1924, and his results were prepared for publication by his friend Pavel S. Alexandroff and published posthumously in 1925 and 1927.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 7 / 29

The Urysohn space U was introdused by Pavel S. Urysohn shortly before his tragic death in 1924, and his results were prepared for publication by his friend Pavel S. Alexandroff and published posthumously in 1925 and 1927. After Urysohn’s result was published, S. Banach and S. Mazur proved that C[0, 1] is universal for all separable metric spaces.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 7 / 29

The Urysohn space U was introdused by Pavel S. Urysohn shortly before his tragic death in 1924, and his results were prepared for publication by his friend Pavel S. Alexandroff and published posthumously in 1925 and 1927. After Urysohn’s result was published, S. Banach and S. Mazur proved that C[0, 1] is universal for all separable metric spaces. But C[0, 1] is NOT ultrahomogeneous.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 7 / 29

The Urysohn space U was introdused by Pavel S. Urysohn shortly before his tragic death in 1924, and his results were prepared for publication by his friend Pavel S. Alexandroff and published posthumously in 1925 and 1927. After Urysohn’s result was published, S. Banach and S. Mazur proved that C[0, 1] is universal for all separable metric spaces. But C[0, 1] is NOT ultrahomogeneous. Dually, ℓ2 is ultrahomogeneous, but it is not universal.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 7 / 29

The Urysohn space U was introdused by Pavel S. Urysohn shortly before his tragic death in 1924, and his results were prepared for publication by his friend Pavel S. Alexandroff and published posthumously in 1925 and 1927. After Urysohn’s result was published, S. Banach and S. Mazur proved that C[0, 1] is universal for all separable metric spaces. But C[0, 1] is NOT ultrahomogeneous. Dually, ℓ2 is ultrahomogeneous, but it is not universal.

Theorem (Berestovsky and Vershik)

The Gromov-Hausdorff distance may be computed by the following formula: dGH(X, Y) = inf{dH

• i(X), j(Y)
• | i : X ֒

→ U, j : Y ֒ → U} where inf is taken over all isometric embeddings i : X ֒ → U and j : Y ֒ → U.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 7 / 29

Denote by Iso U the group of all isometries of U.

Theorem (Gromov)

GH ∼ = 2U/Iso U (an isometry).

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Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 8 / 29

Denote by Iso U the group of all isometries of U.

Theorem (Gromov)

GH ∼ = 2U/Iso U (an isometry). It is a challenging open problem to describe the topological structure of this metric space.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 8 / 29

Denote by Iso U the group of all isometries of U.

Theorem (Gromov)

GH ∼ = 2U/Iso U (an isometry). It is a challenging open problem to describe the topological structure of this metric space. The talk contributes towards this problem.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 8 / 29

Denote by Iso U the group of all isometries of U.

Theorem (Gromov)

GH ∼ = 2U/Iso U (an isometry). It is a challenging open problem to describe the topological structure of this metric space. The talk contributes towards this problem. It is known that GH is a Polish space. Besides, it is easy to see that GH is contractible.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 8 / 29

Denote by Iso U the group of all isometries of U.

Theorem (Gromov)

GH ∼ = 2U/Iso U (an isometry). It is a challenging open problem to describe the topological structure of this metric space. The talk contributes towards this problem. It is known that GH is a Polish space. Besides, it is easy to see that GH is contractible. However it is not known whether GH is an AR? Is GH∼ = ℓ2?

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 8 / 29

In this talk we mainly are interested in the following subspaces of GH denoted by GH(Rn), n ≥ 1 and called the Gromov-Hausdorff hyperspace of Rn.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 9 / 29

In this talk we mainly are interested in the following subspaces of GH denoted by GH(Rn), n ≥ 1 and called the Gromov-Hausdorff hyperspace of Rn. Here GH(Rn) is the subspace of GH consisting of the classes [E] ∈GH whose representative E is a metric subspace of Rn.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 9 / 29

In this talk we mainly are interested in the following subspaces of GH denoted by GH(Rn), n ≥ 1 and called the Gromov-Hausdorff hyperspace of Rn. Here GH(Rn) is the subspace of GH consisting of the classes [E] ∈GH whose representative E is a metric subspace of Rn.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 9 / 29

The Euclidean-Hausdorff distance

For any two compact subsets X, Y which admit an isometric embeddings in a Euclidean space Rn, n ≥ 1, define the Euclidean-Hausdorff distance by the following formula: dEH(X, Y) = inf{dH

• i(X), j(Y)
• | i : X ֒

→ Rn, j : Y ֒ → Rn} where inf is taken over all isometric embeddings i : X ֒ → Rn and j : Y ֒ → Rn.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 10 / 29

The Euclidean-Hausdorff distance

For any two compact subsets X, Y which admit an isometric embeddings in a Euclidean space Rn, n ≥ 1, define the Euclidean-Hausdorff distance by the following formula: dEH(X, Y) = inf{dH

• i(X), j(Y)
• | i : X ֒

→ Rn, j : Y ֒ → Rn} where inf is taken over all isometric embeddings i : X ֒ → Rn and j : Y ֒ → Rn.

Theorem (Well-known)

If X and Y are two isometric subsets of a Euiclidean space Rn, n ≥ 1, then there exists a Euclidean isometry F : Rn → Rn such that F(X) = Y.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 10 / 29

The Euclidean-Hausdorff distance

For any two compact subsets X, Y which admit an isometric embeddings in a Euclidean space Rn, n ≥ 1, define the Euclidean-Hausdorff distance by the following formula: dEH(X, Y) = inf{dH

• i(X), j(Y)
• | i : X ֒

→ Rn, j : Y ֒ → Rn} where inf is taken over all isometric embeddings i : X ֒ → Rn and j : Y ֒ → Rn.

Theorem (Well-known)

If X and Y are two isometric subsets of a Euiclidean space Rn, n ≥ 1, then there exists a Euclidean isometry F : Rn → Rn such that F(X) = Y.

Corollary

If X, Y ⊂ Rn are compact subsets, then dEH(X, Y) = inf{dH

• X, F(Y)
• | F : Rn → Rn is an isometry}
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Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 10 / 29

In other words....

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 11 / 29

In other words.... Denote by E(n) = Iso Rn the group of isometries of Rn.

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Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 11 / 29

In other words.... Denote by E(n) = Iso Rn the group of isometries of Rn. The group E(n) acts continuously on 2Rn: (g, A) → g(A).

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 11 / 29

In other words.... Denote by E(n) = Iso Rn the group of isometries of Rn. The group E(n) acts continuously on 2Rn: (g, A) → g(A). Denote by [X] = {F(X) | F ∈ E(n)} - the orbit of an X ∈ 2Rn. By 2Rn/E(n) we denote the orbit space.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 11 / 29

In other words.... Denote by E(n) = Iso Rn the group of isometries of Rn. The group E(n) acts continuously on 2Rn: (g, A) → g(A). Denote by [X] = {F(X) | F ∈ E(n)} - the orbit of an X ∈ 2Rn. By 2Rn/E(n) we denote the orbit space. Then for [X], [Y] ∈ 2Rn ρ([X], [Y]) = inf{dH

• X, F(Y)
• | F ∈ E(n)}

metrizes the orbit space 2Rn/E(n), and clearly,

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 11 / 29

In other words.... Denote by E(n) = Iso Rn the group of isometries of Rn. The group E(n) acts continuously on 2Rn: (g, A) → g(A). Denote by [X] = {F(X) | F ∈ E(n)} - the orbit of an X ∈ 2Rn. By 2Rn/E(n) we denote the orbit space. Then for [X], [Y] ∈ 2Rn ρ([X], [Y]) = inf{dH

• X, F(Y)
• | F ∈ E(n)}

metrizes the orbit space 2Rn/E(n), and clearly, ρ([X], [Y]) = dEH(X, Y).

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 11 / 29

Main results

Clearly, dGH ≤ dEH. In general dGH(X, Y) may be strictly less than dEH(X, Y). For instance, take X = {a, b, c} - the vertices of an equilateral triangle

• f side lenght 1, and Y = {∗}.
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Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 12 / 29

Main results

Clearly, dGH ≤ dEH. In general dGH(X, Y) may be strictly less than dEH(X, Y). For instance, take X = {a, b, c} - the vertices of an equilateral triangle

• f side lenght 1, and Y = {∗}.

Then dEH(X, Y) =

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 12 / 29

Main results

Clearly, dGH ≤ dEH. In general dGH(X, Y) may be strictly less than dEH(X, Y). For instance, take X = {a, b, c} - the vertices of an equilateral triangle

• f side lenght 1, and Y = {∗}.

Then dEH(X, Y) = √ 3/3

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 12 / 29

Main results

Clearly, dGH ≤ dEH. In general dGH(X, Y) may be strictly less than dEH(X, Y). For instance, take X = {a, b, c} - the vertices of an equilateral triangle

• f side lenght 1, and Y = {∗}.

Then dEH(X, Y) = √ 3/3 while dGH(X, Y) =

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 12 / 29

Main results

Clearly, dGH ≤ dEH. In general dGH(X, Y) may be strictly less than dEH(X, Y). For instance, take X = {a, b, c} - the vertices of an equilateral triangle

• f side lenght 1, and Y = {∗}.

Then dEH(X, Y) = √ 3/3 while dGH(X, Y) = 1/2.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 12 / 29

Main results

Clearly, dGH ≤ dEH. In general dGH(X, Y) may be strictly less than dEH(X, Y). For instance, take X = {a, b, c} - the vertices of an equilateral triangle

• f side lenght 1, and Y = {∗}.

Then dEH(X, Y) = √ 3/3 while dGH(X, Y) = 1/2.

Theorem

GH(Rn) ∼ = 2Rn/E(n).

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 12 / 29

Sketch

2U(Rn) = {A ∈ 2U | ∃ i : A ֒ → Rn}. f : 2Rn → 2U(Rn)/Iso U = GH(Rn), A → [j(A)], where j : A ֒ → U is an embedding.

✲ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❯ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ☛ 2Rn 2Rn/E(n) GH(Rn) = 2U(Rn)/Iso U f p

• f(p(A)) = f(A)
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Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 13 / 29

Since dGH ≤ dEH, we infer that

• f : 2Rn/E(n) → 2U(Rn)/Iso U

is continuous and bijective.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 14 / 29

Since dGH ≤ dEH, we infer that

• f : 2Rn/E(n) → 2U(Rn)/Iso U

is continuous and bijective. For continuity of the inverse map

• f −1 : 2U(Rn)/Iso U → 2Rn/E(n)

we use the following

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 14 / 29

Since dGH ≤ dEH, we infer that

• f : 2Rn/E(n) → 2U(Rn)/Iso U

is continuous and bijective. For continuity of the inverse map

• f −1 : 2U(Rn)/Iso U → 2Rn/E(n)

we use the following

Theorem (Memoli)

dEH ≤ Cn ·

• dGH,
• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 14 / 29

Since dGH ≤ dEH, we infer that

• f : 2Rn/E(n) → 2U(Rn)/Iso U

is continuous and bijective. For continuity of the inverse map

• f −1 : 2U(Rn)/Iso U → 2Rn/E(n)

we use the following

Theorem (Memoli)

dEH ≤ Cn ·

• dGH,

Thus

• f : 2Rn/E(n) → 2U(Rn)/Iso U

is a homeomorphism: GH(Rn) ∼ = 2Rn/E(n).

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 14 / 29

Theorem

The action E(n) 2Rn is proper.

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Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 15 / 29

Theorem

The action E(n) 2Rn is proper. Here proper means that for any compact subset K ⊂ 2Rn, the transporter K, K = {g ∈ E(n) | gK ∩ K = ∅} has compact closure in E(n).

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 15 / 29

Theorem

The action E(n) 2Rn is proper. Here proper means that for any compact subset K ⊂ 2Rn, the transporter K, K = {g ∈ E(n) | gK ∩ K = ∅} has compact closure in E(n).

Facts

In a proper G-space each stabilizer Gx = {g ∈ G | gx = x} is compact. Every obit G(x) is closed and G(x) ∼ =G G/Gx,

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 15 / 29

Definition

Let G be a locally compact group and H ⊂ G a compact subgroup. Then a subset S ⊂ X of a proper G-space X is a global H-slice of X, if

1

S is H-invariant,

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Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 16 / 29

Definition

Let G be a locally compact group and H ⊂ G a compact subgroup. Then a subset S ⊂ X of a proper G-space X is a global H-slice of X, if

1

S is H-invariant,

2

S is closed in X,

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 16 / 29

Definition

Let G be a locally compact group and H ⊂ G a compact subgroup. Then a subset S ⊂ X of a proper G-space X is a global H-slice of X, if

1

S is H-invariant,

2

S is closed in X,

3

if gS ∩ S = ∅ then g ∈ H

4

G(S) :=

g∈G

gS = X.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 16 / 29

Definition

Let G be a locally compact group and H ⊂ G a compact subgroup. Then a subset S ⊂ X of a proper G-space X is a global H-slice of X, if

1

S is H-invariant,

2

S is closed in X,

3

if gS ∩ S = ∅ then g ∈ H

4

G(S) :=

g∈G

gS = X.

G(S) S gS

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 16 / 29

The Chebyshev balls

Theorem (P .L. Chebyshev)

For every compact subset A ⊂ Rn, there is a unique closed ball Ch(A), called the Chebyshev ball of A, such that A ⊂ Ch(A).

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 17 / 29

The Chebyshev balls

Theorem (P .L. Chebyshev)

For every compact subset A ⊂ Rn, there is a unique closed ball Ch(A), called the Chebyshev ball of A, such that A ⊂ Ch(A).

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 17 / 29

• If Ch(A) = B(b, r), then we denote ch(A) = b – the Chebyshev

center of A; it belongs to conv A.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 18 / 29

• If Ch(A) = B(b, r), then we denote ch(A) = b – the Chebyshev

center of A; it belongs to conv A.

• R(A) = r – the Chebyshev radius of A.
• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 18 / 29

• If Ch(A) = B(b, r), then we denote ch(A) = b – the Chebyshev

center of A; it belongs to conv A.

• R(A) = r – the Chebyshev radius of A.

Theorem

ch : 2Rn → Rn is an E(n)-equivariant map, i.e., ch(gA) = gch(A), A ∈ 2Rn, g ∈ E(n).

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 18 / 29

• If Ch(A) = B(b, r), then we denote ch(A) = b – the Chebyshev

center of A; it belongs to conv A.

• R(A) = r – the Chebyshev radius of A.

Theorem

ch : 2Rn → Rn is an E(n)-equivariant map, i.e., ch(gA) = gch(A), A ∈ 2Rn, g ∈ E(n).

Corollary

The inverse image T(Rn) := ch−1(0) is a global O(n)-slice for 2Rn.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 18 / 29

• If Ch(A) = B(b, r), then we denote ch(A) = b – the Chebyshev

center of A; it belongs to conv A.

• R(A) = r – the Chebyshev radius of A.

Theorem

ch : 2Rn → Rn is an E(n)-equivariant map, i.e., ch(gA) = gch(A), A ∈ 2Rn, g ∈ E(n).

Corollary

The inverse image T(Rn) := ch−1(0) is a global O(n)-slice for 2Rn.

Theorem

GH(Rn) = 2Rn/E(n) ∼ = T(Rn)/O(n).

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 18 / 29

How to compute T(Rn)/O(n).? Denote Ch(n) := {A ∈ 2Rn | Ch(A) = Bn}.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 19 / 29

How to compute T(Rn)/O(n).? Denote Ch(n) := {A ∈ 2Rn | Ch(A) = Bn}.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 19 / 29

How to compute T(Rn)/O(n).? Denote Ch(n) := {A ∈ 2Rn | Ch(A) = Bn}.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 19 / 29

How to compute T(Rn)/O(n).? Denote Ch(n) := {A ∈ 2Rn | Ch(A) = Bn}.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 19 / 29

Proposition

1

T(Rn) ∼ =O(n) OCone

• Ch(n)
• .

2

T(Rn)/O(n) ∼ = OCone

• Ch(n)/O(n)
• .
• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 20 / 29

Proposition

1

T(Rn) ∼ =O(n) OCone

• Ch(n)
• .

2

T(Rn)/O(n) ∼ = OCone

• Ch(n)/O(n)
• .

Recall the definition of an open cone: OCone(X) = X × [0, ∞)/X × {0}.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 20 / 29

Proposition

1

T(Rn) ∼ =O(n) OCone

• Ch(n)
• .

2

T(Rn)/O(n) ∼ = OCone

• Ch(n)/O(n)
• .

Recall the definition of an open cone: OCone(X) = X × [0, ∞)/X × {0}. Proof. f(A) =

• 1

R(A) · A,

if R(A) = 0 θ, if A = {0}.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 20 / 29

Orbit spaces of Hyperspaces

How to compute the orbit space Ch(n)/O(n)?

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 21 / 29

Orbit spaces of Hyperspaces

How to compute the orbit space Ch(n)/O(n)? Recall that Ch(n) := {A ∈ 2Bn | Ch(A) = Bn} ⊂ 2Bn.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 21 / 29

Orbit spaces of Hyperspaces

How to compute the orbit space Ch(n)/O(n)? Recall that Ch(n) := {A ∈ 2Bn | Ch(A) = Bn} ⊂ 2Bn. Ch(n)/O(n) ⊂ 2Bn/O(n).

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 21 / 29

Orbit spaces of Hyperspaces

How to compute the orbit space Ch(n)/O(n)? Recall that Ch(n) := {A ∈ 2Bn | Ch(A) = Bn} ⊂ 2Bn. Ch(n)/O(n) ⊂ 2Bn/O(n). It is in order to mention that studying orbit spaces of hyperspaces goes back to Jim West (1976) (see [5], [6]).

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 21 / 29

Orbit spaces of Hyperspaces

How to compute the orbit space Ch(n)/O(n)? Recall that Ch(n) := {A ∈ 2Bn | Ch(A) = Bn} ⊂ 2Bn. Ch(n)/O(n) ⊂ 2Bn/O(n). It is in order to mention that studying orbit spaces of hyperspaces goes back to Jim West (1976) (see [5], [6]).

Theorem (H. Toru´ nczyk and J. West, 1980)

2SO(2)/SO(2) ∈ AR and 2SO(2)/SO(2) ≇ Q

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 21 / 29

Orbit spaces of Hyperspaces

How to compute the orbit space Ch(n)/O(n)? Recall that Ch(n) := {A ∈ 2Bn | Ch(A) = Bn} ⊂ 2Bn. Ch(n)/O(n) ⊂ 2Bn/O(n). It is in order to mention that studying orbit spaces of hyperspaces goes back to Jim West (1976) (see [5], [6]).

Theorem (H. Toru´ nczyk and J. West, 1980)

2SO(2)/SO(2) ∈ AR and 2SO(2)/SO(2) ≇ Q

Theorem (S.A., 2000)

2SO(2)/O(2) ∈ AR and 2SO(2)/O(2) ∼ = BM(2) ≇ Q

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 21 / 29

Properties of Ch(n)

Theorem

1

Ch(n) is O(n)-AR.

2

Ch(n)/O(n) is an AR.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 22 / 29

Properties of Ch(n)

Theorem

1

Ch(n) is O(n)-AR.

2

Ch(n)/O(n) is an AR.

Theorem

1

Ch(n) ∼ = Q := [0, 1]∞.

2

Ch(n)/O(n) ∼ = Q.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 22 / 29

Theorem

GH(Rn) = 2Rn/E(n) ∼ = Q \ {∗}.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 23 / 29

Theorem

GH(Rn) = 2Rn/E(n) ∼ = Q \ {∗}.

Proof.

One has 2Rn/E(n) ∼ = OCone

• Ch(n)/O(n)

∼ = OCone

• Q
• .
• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 23 / 29

Theorem

GH(Rn) = 2Rn/E(n) ∼ = Q \ {∗}.

Proof.

One has 2Rn/E(n) ∼ = OCone

• Ch(n)/O(n)

∼ = OCone

• Q
• .

But, it is well known (T.A. Chapman) that the open cone OCone(Q) ∼ = Q \ {∗}.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 23 / 29

[1] S.A. Antonyan, West’s problem on equivariant hyperspaces and Banach-Mazur compacta, Trans. Amer. Math. Soc. 355,

• no. 8 (2003), 3379-3404.

[2]

• M. Gromov. Metric structures for Riemannian and

non-Riemannian spaces, Progress in Mathematics 152, Birkh¨ auser (1999). [3]

• M. Gromov. Structures m´

etriques pour les vari´ et´ es

• riemanniennes. Vol. 1 of Textes Math´

ematiques [Mathematical Texts], CEDIC, Paris, 1981. Edited by J. Lafontaine and P . Pansu. [4] T.A. Chapman, Lectures on Hilbert cube manifolds, C. B. M. S. Regional Conference Series in Math., 28, Amer. Math. Soc., 1976. [5] J.E. West, Induced involutions on Hilbert cube hyperspaces, Topology Proc. 1 (1976), 281- 293.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 23 / 29

[6]

• H. Toru´

nczyk and J.E. West, The fine structure of S1/S1; a Q-manifold hyperspace localiza- tion of the integers, in: Proc.

• Internat. Conf. Geom. Topol., 439-449, PWN-Pol. Sci. Publ.,

Warszawa, 1980.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 24 / 29

THE END !

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 24 / 29

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 25 / 29

Some ideas of the proof that Ch(n)/O(n) ∼ = Q.

Theorem (H. Toru´ nczyk, 1978)

A a compact metrizable space X is homeomorphic to the Hilbert cube iff

• X is an AR.
• X satisfies the DDP (Disjoint Discs Property).
• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 25 / 29

Some ideas of the proof that Ch(n)/O(n) ∼ = Q.

Theorem (H. Toru´ nczyk, 1978)

A a compact metrizable space X is homeomorphic to the Hilbert cube iff

• X is an AR.
• X satisfies the DDP (Disjoint Discs Property).

Theorem (S. Antonyan, 1988)

Let G be a compact group, X a metrizable G-AR. Then the orbit space X/G is an AR.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 25 / 29

DDnP and DDP

Definition

Y satisfies DDnP for a given integer n ≥ 0, if each map f : Bn → Y can be arbitrary closely approximated by two maps f1, f2 : Bn → Y with disjoint images.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 26 / 29

DDnP and DDP

Definition

Y satisfies DDnP for a given integer n ≥ 0, if each map f : Bn → Y can be arbitrary closely approximated by two maps f1, f2 : Bn → Y with disjoint images. Y satisfies DDP , if it satisfies DDnP for all n ≥ 0.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 26 / 29

DDnP and DDP

Definition

Y satisfies DDnP for a given integer n ≥ 0, if each map f : Bn → Y can be arbitrary closely approximated by two maps f1, f2 : Bn → Y with disjoint images. Y satisfies DDP , if it satisfies DDnP for all n ≥ 0.

Proposition

A compact metric ANR space X satisfies the property DDP iff for every ε > 0, there exist two continuous maps fε, gε : X → X such that:

1

ρ(x, fε(x)) < ε and ρ(x, gε(x)) < ε for all x ∈ X.

2

Im fε ∩ Im gε = ∅.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 26 / 29

Equivariant DDP

Theorem

For every ε > 0, there exist two continuous O(n)-equivariant maps fε, gε : Ch(n) → Ch(n) such that:

1

ρ

• A, fε(A)
• < ε and ρ
• A, gε(A)
• < ε for all A ∈ Ch(n).

2

Im fε ∩ Im gε = ∅.

• First we define the map fε : Ch(n) → Ch(n) by
• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 27 / 29

Equivariant DDP

Theorem

For every ε > 0, there exist two continuous O(n)-equivariant maps fε, gε : Ch(n) → Ch(n) such that:

1

ρ

• A, fε(A)
• < ε and ρ
• A, gε(A)
• < ε for all A ∈ Ch(n).

2

Im fε ∩ Im gε = ∅.

• First we define the map fε : Ch(n) → Ch(n) by

fε(A) = {x ∈ Bn | dist(x, A) ≤ ε}.

f (A)

A

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 27 / 29

Equivariant DDP

Theorem

For every ε > 0, there exist two continuous O(n)-equivariant maps fε, gε : Ch(n) → Ch(n) such that:

1

ρ

• A, fε(A)
• < ε and ρ
• A, gε(A)
• < ε for all A ∈ Ch(n).

2

Im fε ∩ Im gε = ∅.

• First we define the map fε : Ch(n) → Ch(n) by

fε(A) = {x ∈ Bn | dist(x, A) ≤ ε}.

f (A)

A

It is clear that fε(A) ∈ Ch(n) whenever A ∈ Ch(n).

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 27 / 29

Proof.

• The second map

gε : Ch(n) → Ch(n) is defined in such a way that Int gε(A) = ∅. Then Im fε ∩ Im gε = ∅.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 28 / 29

Proof.

• The second map

gε : Ch(n) → Ch(n) is defined in such a way that Int gε(A) = ∅. Then Im fε ∩ Im gε = ∅. Then the maps fε, gε : Ch(n) → Ch(n) induce

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 28 / 29

Proof.

• The second map

gε : Ch(n) → Ch(n) is defined in such a way that Int gε(A) = ∅. Then Im fε ∩ Im gε = ∅. Then the maps fε, gε : Ch(n) → Ch(n) induce

• fε,

gε : Ch(n)/O(n) → Ch(n)/O(n) which are ε-close to the indentity map and have disjoint images.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 28 / 29

[1] S.A. Antonyan, West’s problem on equivariant hyperspaces and Banach-Mazur compacta, Trans. Amer. Math. Soc. 355,

• no. 8 (2003), 3379-3404.

[2]

• M. Gromov. Metric structures for Riemannian and

non-Riemannian spaces, Progress in Mathematics 152, Birkh¨ auser (1999). [3]

• M. Gromov. Structures m´

etriques pour les vari´ et´ es

• riemanniennes. Vol. 1 of Textes Math´

ematiques [Mathematical Texts], CEDIC, Paris, 1981. Edited by J. Lafontaine and P . Pansu. [4] T.A. Chapman, Lectures on Hilbert cube manifolds, C. B. M. S. Regional Conference Series in Math., 28, Amer. Math. Soc., 1976. [5] J.E. West, Induced involutions on Hilbert cube hyperspaces, Topology Proc. 1 (1976), 281- 293.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 28 / 29

[6]

• H. Toru´

nczyk and J.E. West, The fine structure of S1/S1; a Q-manifold hyperspace localiza- tion of the integers, in: Proc.

• Internat. Conf. Geom. Topol., 439-449, PWN-Pol. Sci. Publ.,

Warszawa, 1980.

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 29 / 29

THE END !

• S. Antonyan (UNAM)

Hyperspaces in the Gromov-Haisdorff metric July 25-29, 2016 29 / 29