HYPERSPACES OF COMPACT CONVEX SETS Sergey Antonyan and Natalia - PowerPoint PPT Presentation

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva � V x , V y � = { g ∈ G | gV x ∩ V y � = ∅} y V y V x gV x x HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Global Slices Definition Let X be a G -space, and H ≤ G a closed subgroup. A subset S ⊂ X is called a global H -slice if the following conditions hold: G ( S ) = X , where G ( S ) = � g ∈ G gS . S is closed in G ( S ). S is H -invariant. gS ∩ S = ∅ for all g ∈ G \ H . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Global Slices Definition Let X be a G -space, and H ≤ G a closed subgroup. A subset S ⊂ X is called a global H -slice if the following conditions hold: G ( S ) = X , where G ( S ) = � g ∈ G gS . S is closed in G ( S ). S is H -invariant. gS ∩ S = ∅ for all g ∈ G \ H . GS S HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Global Slices Definition Let X be a G -space, and H ≤ G a closed subgroup. A subset S ⊂ X is called a global H -slice if the following conditions hold: G ( S ) = X , where G ( S ) = � g ∈ G gS . S is closed in G ( S ). S is H -invariant. gS ∩ S = ∅ for all g ∈ G \ H . GS S HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Global Slices Definition Let X be a G -space, and H ≤ G a closed subgroup. A subset S ⊂ X is called a global H -slice if the following conditions hold: G ( S ) = X , where G ( S ) = � g ∈ G gS . S is closed in G ( S ). S is H -invariant. gS ∩ S = ∅ for all g ∈ G \ H . HS=S GS S HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Global Slices Definition Let X be a G -space, and H ≤ G a closed subgroup. A subset S ⊂ X is called a global H -slice if the following conditions hold: G ( S ) = X , where G ( S ) = � g ∈ G gS . S is closed in G ( S ). S is H -invariant. gS ∩ S = ∅ for all g ∈ G \ H . GS S gS HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Theorem (Palais, 1961) Let G be a Lie group, X be a proper G-space and x ∈ X. Then there exists a G-invariant neighborhood U of x which admits a global G x -slice S for U. Equivalent form: there exists a G -map f − 1 ( eG x ) = S . f : U → G / G x such that If, in addition, G is a Lie group having finitely many connected components, then a maximal compact subgroup K ⊂ G exists. In this case gG x g − 1 ⊂ K , and hence, there is a G -map q : G / G x → G / K . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Theorem (Palais, 1961) Let G be a Lie group, X be a proper G-space and x ∈ X. Then there exists a G-invariant neighborhood U of x which admits a global G x -slice S for U. Equivalent form: there exists a G -map f − 1 ( eG x ) = S . f : U → G / G x such that If, in addition, G is a Lie group having finitely many connected components, then a maximal compact subgroup K ⊂ G exists. In this case gG x g − 1 ⊂ K , and hence, there is a G -map q : G / G x → G / K . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Theorem (Palais, 1961) Let G be a Lie group, X be a proper G-space and x ∈ X. Then there exists a G-invariant neighborhood U of x which admits a global G x -slice S for U. Equivalent form: there exists a G -map f − 1 ( eG x ) = S . f : U → G / G x such that If, in addition, G is a Lie group having finitely many connected components, then a maximal compact subgroup K ⊂ G exists. In this case gG x g − 1 ⊂ K , and hence, there is a G -map q : G / G x → G / K . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Consider the composition: q f G/G G/K U x F The inverse image Q = F − 1 ( eK ) is a K -slice for U . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Consider the composition: q f G/K G/G U x F The inverse image Q = F − 1 ( eK ) is a K -slice for U . GS S HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Consider the composition: q f G/K G/G U x F The inverse image Q = F − 1 ( eK ) is a K -slice for U . GS S HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Consider the composition: q f G/K G/G U x F The inverse image Q = F − 1 ( eK ) is a K -slice for U . HS=S GS S HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Consider the composition: q f G/K G/G U x F The inverse image Q = F − 1 ( eK ) is a K -slice for U . GS S gS HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva This K -slices can be pasted together to obtain a global K -slice of X . Theorem (Abels, 1974) Let G be a Lie group having finitely many connected components, K a maximal compact subgroup and X a proper G-space. If the orbit space X / G is paracompact then (1) X admits a global K-slice S. (2) X is K-equivariantly homeomorphic to the product S × G / K. HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva This K -slices can be pasted together to obtain a global K -slice of X . Theorem (Abels, 1974) Let G be a Lie group having finitely many connected components, K a maximal compact subgroup and X a proper G-space. If the orbit space X / G is paracompact then (1) X admits a global K-slice S. (2) X is K-equivariantly homeomorphic to the product S × G / K. HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Aff( n ) has two connected components. O ( n ), the orthogonal group, is a maximal compact subgroup of Aff( n ). Aff( n ) acts properly on cb ( R n ). The orbit space cb ( R n ) / Aff( n ) is metrizable and compact. Hence, there exists a global O ( n )-slice S for cb ( R n ) and cb ( R n ) ∼ = S × Aff ( n ) / O ( n ) . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Aff( n ) has two connected components. O ( n ), the orthogonal group, is a maximal compact subgroup of Aff( n ). Aff( n ) acts properly on cb ( R n ). The orbit space cb ( R n ) / Aff( n ) is metrizable and compact. Hence, there exists a global O ( n )-slice S for cb ( R n ) and cb ( R n ) ∼ = S × Aff ( n ) / O ( n ) . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva = R n × GL ( n ) / O ( n ) . Aff( n ) / O ( n ) ∼ RQ -Decomposition Theorem Every invertible matrix can be uniquely represented as the product of an orthogonal matrix and an upper triangular matrix with positive elements in the diagonal. GL ( n ) / O ( n ) is homeomorphic to R n ( n +1) / 2 = R n × R n ( n +1) / 2 = R n ( n +3) / 2 . Aff( n ) / O ( n ) ∼ cb ( R n ) ∼ = S × R n ( n +3) / 2 . It remains to find S . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva = R n × GL ( n ) / O ( n ) . Aff( n ) / O ( n ) ∼ RQ -Decomposition Theorem Every invertible matrix can be uniquely represented as the product of an orthogonal matrix and an upper triangular matrix with positive elements in the diagonal. GL ( n ) / O ( n ) is homeomorphic to R n ( n +1) / 2 = R n × R n ( n +1) / 2 = R n ( n +3) / 2 . Aff( n ) / O ( n ) ∼ cb ( R n ) ∼ = S × R n ( n +3) / 2 . It remains to find S . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva = R n × GL ( n ) / O ( n ) . Aff( n ) / O ( n ) ∼ RQ -Decomposition Theorem Every invertible matrix can be uniquely represented as the product of an orthogonal matrix and an upper triangular matrix with positive elements in the diagonal. GL ( n ) / O ( n ) is homeomorphic to R n ( n +1) / 2 = R n × R n ( n +1) / 2 = R n ( n +3) / 2 . Aff( n ) / O ( n ) ∼ cb ( R n ) ∼ = S × R n ( n +3) / 2 . It remains to find S . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva = R n × GL ( n ) / O ( n ) . Aff( n ) / O ( n ) ∼ RQ -Decomposition Theorem Every invertible matrix can be uniquely represented as the product of an orthogonal matrix and an upper triangular matrix with positive elements in the diagonal. GL ( n ) / O ( n ) is homeomorphic to R n ( n +1) / 2 = R n × R n ( n +1) / 2 = R n ( n +3) / 2 . Aff( n ) / O ( n ) ∼ cb ( R n ) ∼ = S × R n ( n +3) / 2 . It remains to find S . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva = R n × GL ( n ) / O ( n ) . Aff( n ) / O ( n ) ∼ RQ -Decomposition Theorem Every invertible matrix can be uniquely represented as the product of an orthogonal matrix and an upper triangular matrix with positive elements in the diagonal. GL ( n ) / O ( n ) is homeomorphic to R n ( n +1) / 2 = R n × R n ( n +1) / 2 = R n ( n +3) / 2 . Aff( n ) / O ( n ) ∼ cb ( R n ) ∼ = S × R n ( n +3) / 2 . It remains to find S . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva = R n × GL ( n ) / O ( n ) . Aff( n ) / O ( n ) ∼ RQ -Decomposition Theorem Every invertible matrix can be uniquely represented as the product of an orthogonal matrix and an upper triangular matrix with positive elements in the diagonal. GL ( n ) / O ( n ) is homeomorphic to R n ( n +1) / 2 = R n × R n ( n +1) / 2 = R n ( n +3) / 2 . Aff( n ) / O ( n ) ∼ cb ( R n ) ∼ = S × R n ( n +3) / 2 . It remains to find S . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva The John ellipsoid For every compact convex body A ∈ cb ( R n ) there exists a unique minimal volume ellipsoid j ( A ) containing A . The ellipsoid j ( A ) is called the John (sometimes also the L¨ owner) ellipsoid of A . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva The John ellipsoid For every compact convex body A ∈ cb ( R n ) there exists a unique minimal volume ellipsoid j ( A ) containing A . The ellipsoid j ( A ) is called the John (sometimes also the L¨ owner) ellipsoid of A . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva The map � B n � j : cb ( R n ) → E ( n ) = Aff( n ) ⊂ cb ( R n ) is Aff( n )-equivariant, i.e., g ∈ Aff( n ) , and A ∈ cb ( R n ) . j ( gA ) = gj ( A ) for every A HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva The map � B n � j : cb ( R n ) → E ( n ) = Aff( n ) ⊂ cb ( R n ) is Aff( n )-equivariant, i.e., g ∈ Aff( n ) , and A ∈ cb ( R n ) . j ( gA ) = gj ( A ) for every g A HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva The map � B n � j : cb ( R n ) → E ( n ) = Aff( n ) ⊂ cb ( R n ) is Aff( n )-equivariant, i.e., g ∈ Aff( n ) , and A ∈ cb ( R n ) . j ( gA ) = gj ( A ) for every g gA A HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva The map � B n � j : cb ( R n ) → E ( n ) = Aff( n ) ⊂ cb ( R n ) is Aff( n )-equivariant, i.e., g ∈ Aff( n ) , and A ∈ cb ( R n ) . j ( gA ) = gj ( A ) for every g gA A HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva The map � B n � j : cb ( R n ) → E ( n ) = Aff( n ) ⊂ cb ( R n ) is Aff( n )-equivariant, i.e., g ∈ Aff( n ) , and A ∈ cb ( R n ) . j ( gA ) = gj ( A ) for every g gA A HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva For every A ∈ cb ( R n ) there exists an affine transformation g ∈ Aff( n ) such that j ( A ) = g B n where B n is the closed Euclidean unit ball. A j ( g − 1 A ) = g − 1 j ( A ) = B n . Thus, HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva For every A ∈ cb ( R n ) there exists an affine transformation g ∈ Aff( n ) such that j ( A ) = g B n where B n is the closed Euclidean unit ball. A j ( g − 1 A ) = g − 1 j ( A ) = B n . Thus, HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva For every A ∈ cb ( R n ) there exists an affine transformation g ∈ Aff( n ) such that j ( A ) = g B n where B n is the closed Euclidean unit ball. A g j ( g − 1 A ) = g − 1 j ( A ) = B n . Thus, HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva For every A ∈ cb ( R n ) there exists an affine transformation g ∈ Aff( n ) such that j ( A ) = g B n where B n is the closed Euclidean unit ball. A g j ( g − 1 A ) = g − 1 j ( A ) = B n . Thus, HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva For every A ∈ cb ( R n ) there exists an affine transformation g ∈ Aff( n ) such that j ( A ) = g B n where B n is the closed Euclidean unit ball. g -1 A g j ( g − 1 A ) = g − 1 j ( A ) = B n . Thus, HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva For every A ∈ cb ( R n ) there exists an affine transformation g ∈ Aff( n ) such that j ( A ) = g B n where B n is the closed Euclidean unit ball. g -1 g -1 A A g j ( g − 1 A ) = g − 1 j ( A ) = B n . Thus, HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva For every n ≥ 2 lets denote by J ( n ) the following set: J ( n ) = { A ∈ cb ( R n ) | j ( A ) = B n } . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva For every n ≥ 2 lets denote by J ( n ) the following set: J ( n ) = { A ∈ cb ( R n ) | j ( A ) = B n } . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva For every n ≥ 2 lets denote by J ( n ) the following set: J ( n ) = { A ∈ cb ( R n ) | j ( A ) = B n } . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva For every n ≥ 2 lets denote by J ( n ) the following set: J ( n ) = { A ∈ cb ( R n ) | j ( A ) = B n } . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva 1 J ( n ) is O ( n )-invariant, 2 Aff( n )( J ( n )) = cb ( R n ), 3 J ( n ) is closed in cb ( R n ), 4 If A ∈ J ( n ) and g ∈ Aff( n ) \ O ( n ) then B n � = g B n = gj ( A ) = j ( gA ) HYPERSPACES OF COMPACT CONVEX SETS and hence J ( n ) ∩ gJ ( n ) = ∅ .

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva 1 J ( n ) is O ( n )-invariant, 2 Aff( n )( J ( n )) = cb ( R n ), 3 J ( n ) is closed in cb ( R n ), 4 If A ∈ J ( n ) and g ∈ Aff( n ) \ O ( n ) then B n � = g B n = gj ( A ) = j ( gA ) HYPERSPACES OF COMPACT CONVEX SETS and hence J ( n ) ∩ gJ ( n ) = ∅ .

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva 1 J ( n ) is O ( n )-invariant, 2 Aff( n )( J ( n )) = cb ( R n ), 3 J ( n ) is closed in cb ( R n ), 4 If A ∈ J ( n ) and g ∈ Aff( n ) \ O ( n ) then B n � = g B n = gj ( A ) = j ( gA ) HYPERSPACES OF COMPACT CONVEX SETS and hence J ( n ) ∩ gJ ( n ) = ∅ .

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva 1 J ( n ) is O ( n )-invariant, 2 Aff( n )( J ( n )) = cb ( R n ), 3 J ( n ) is closed in cb ( R n ), 4 If A ∈ J ( n ) and g ∈ Aff( n ) \ O ( n ) then B n � = g B n = gj ( A ) = j ( gA ) and hence J ( n ) ∩ gJ ( n ) = ∅ . Theorem J ( n ) is a global O ( n ) -slice for cb ( R n ) . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva 1 J ( n ) is O ( n )-invariant, 2 Aff( n )( J ( n )) = cb ( R n ), 3 J ( n ) is closed in cb ( R n ), 4 If A ∈ J ( n ) and g ∈ Aff( n ) \ O ( n ) then B n � = g B n = gj ( A ) = j ( gA ) and hence J ( n ) ∩ gJ ( n ) = ∅ . Theorem J ( n ) is a global O ( n ) -slice for cb ( R n ) . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva 1 J ( n ) is O ( n )-invariant, 2 Aff( n )( J ( n )) = cb ( R n ), 3 J ( n ) is closed in cb ( R n ), 4 If A ∈ J ( n ) and g ∈ Aff( n ) \ O ( n ) then B n � = g B n = gj ( A ) = j ( gA ) and hence J ( n ) ∩ gJ ( n ) = ∅ . Theorem J ( n ) is a global O ( n ) -slice for cb ( R n ) . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva 1 J ( n ) is O ( n )-invariant, 2 Aff( n )( J ( n )) = cb ( R n ), 3 J ( n ) is closed in cb ( R n ), 4 If A ∈ J ( n ) and g ∈ Aff( n ) \ O ( n ) then B n � = g B n = gj ( A ) = j ( gA ) and hence J ( n ) ∩ gJ ( n ) = ∅ . Theorem J ( n ) is a global O ( n ) -slice for cb ( R n ) . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Hence, cb ( R n ) ∼ = J ( n ) × R n ( n +3) / 2 . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Computing J ( n ) Theorem J ( n ) is an O ( n ) - AR (and hence, it is an AR ). Proof. Being a global O ( n )-slice, J ( n ) is an O ( n )-retract of cb ( R n ). But cb ( R n ) ∈ O ( n )-AR since k � Λ k ( A 1 , . . . A k , t 1 , . . . t k ) = t i A i , k = 1 , 2 , . . . i =1 defines an O ( n )-equivariant convex structure on cb ( R n ). HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva We will show that J ( n ) is a Hilbert cube. Theorem The singleton { B n } is a Z-set in J ( n ) . Moreover, if K ⊂ O ( n ) is a closed subgroup that acts nontransitively on the sphere S n − 1 , then for every ε > 0 , there exists a K-map, χ ε : J ( n ) → J 0 ( n ) , ε -close to the identity map of J ( n ) . Lets denote by J 0 ( n ) = J ( n ) \ { B n } . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva We will show that J ( n ) is a Hilbert cube. Theorem The singleton { B n } is a Z-set in J ( n ) . Moreover, if K ⊂ O ( n ) is a closed subgroup that acts nontransitively on the sphere S n − 1 , then for every ε > 0 , there exists a K-map, χ ε : J ( n ) → J 0 ( n ) , ε -close to the identity map of J ( n ) . Lets denote by J 0 ( n ) = J ( n ) \ { B n } . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva We will show that J ( n ) is a Hilbert cube. Theorem The singleton { B n } is a Z-set in J ( n ) . Moreover, if K ⊂ O ( n ) is a closed subgroup that acts nontransitively on the sphere S n − 1 , then for every ε > 0 , there exists a K-map, χ ε : J ( n ) → J 0 ( n ) , ε -close to the identity map of J ( n ) . Lets denote by J 0 ( n ) = J ( n ) \ { B n } . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva We will show that J ( n ) is a Hilbert cube. Theorem The singleton { B n } is a Z-set in J ( n ) . Moreover, if K ⊂ O ( n ) is a closed subgroup that acts nontransitively on the sphere S n − 1 , then for every ε > 0 , there exists a K-map, χ ε : J ( n ) → J 0 ( n ) , ε -close to the identity map of J ( n ) . Lets denote by J 0 ( n ) = J ( n ) \ { B n } . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Theorem J 0 ( n ) satisfies the equivariant DDP: for every ε > 0 , there exist O ( n ) -maps, f ε , h ε : J 0 ( n ) → J 0 ( n ) , ε -close to the identity map of J 0 ( n ) and such that Im f ε ∩ Im h ε = ∅ . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Theorem J 0 ( n ) satisfies the equivariant DDP: for every ε > 0 , there exist O ( n ) -maps, f ε , h ε : J 0 ( n ) → J 0 ( n ) , ε -close to the identity map of J 0 ( n ) and such that Im f ε ∩ Im h ε = ∅ . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Theorem J 0 ( n ) satisfies the equivariant DDP: for every ε > 0 , there exist O ( n ) -maps, f ε , h ε : J 0 ( n ) → J 0 ( n ) , ε -close to the identity map of J 0 ( n ) and such that Im f ε ∩ Im h ε = ∅ . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Theorem J 0 ( n ) satisfies the equivariant DDP: for every ε > 0 , there exist O ( n ) -maps, f ε , h ε : J 0 ( n ) → J 0 ( n ) , ε -close to the identity map of J 0 ( n ) and such that Im f ε ∩ Im h ε = ∅ . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Theorem J 0 ( n ) satisfies the equivariant DDP: for every ε > 0 , there exist O ( n ) -maps, f ε , h ε : J 0 ( n ) → J 0 ( n ) , ε -close to the identity map of J 0 ( n ) and such that Im f ε ∩ Im h ε = ∅ . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Theorem J 0 ( n ) satisfies the equivariant DDP: for every ε > 0 , there exist O ( n ) -maps, f ε , h ε : J 0 ( n ) → J 0 ( n ) , ε -close to the identity map of J 0 ( n ) and such that Im f ε ∩ Im h ε = ∅ . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva According to Toru´ nczyk’s Characterization Theorem, we have: Corollary J 0 ( n ) is a Q-manifold and hence J ( n ) is a Hilbert cube. Corollary cb ( R n ) is homeomorphic to Q × R n ( n +3) / 2 . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva According to Toru´ nczyk’s Characterization Theorem, we have: Corollary J 0 ( n ) is a Q-manifold and hence J ( n ) is a Hilbert cube. Corollary cb ( R n ) is homeomorphic to Q × R n ( n +3) / 2 . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva According to Toru´ nczyk’s Characterization Theorem, we have: Corollary J 0 ( n ) is a Q-manifold and hence J ( n ) is a Hilbert cube. Corollary cb ( R n ) is homeomorphic to Q × R n ( n +3) / 2 . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva X H = { x ∈ X | hx = x , ∀ h ∈ H } Corollary (c) for a closed subgroup H ⊂ O ( n ) that acts nontransitively on S n − 1 , the H-fixed point set J ( n ) H is homeomorphic to the Hilbert cube. (d) for a closed subgroup H ⊂ O ( n ) that acts nontransitively on S n − 1 , the H-orbit space J ( n ) / H is homeomorphic to the Hilbert cube. (e) for any closed subgroup H ⊂ O ( n ) , the H-orbit space J 0 ( n ) / H is a Q-manifold. HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva X H = { x ∈ X | hx = x , ∀ h ∈ H } Corollary (c) for a closed subgroup H ⊂ O ( n ) that acts nontransitively on S n − 1 , the H-fixed point set J ( n ) H is homeomorphic to the Hilbert cube. (d) for a closed subgroup H ⊂ O ( n ) that acts nontransitively on S n − 1 , the H-orbit space J ( n ) / H is homeomorphic to the Hilbert cube. (e) for any closed subgroup H ⊂ O ( n ) , the H-orbit space J 0 ( n ) / H is a Q-manifold. HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva The Banach-Mazur compacta In his 1932 book Th´ eorie des Op´ erations Lin´ eaires , S. Banach introduced the space of isometry classes [ X ], of n -dimensional Banach spaces X equipped with the well-known Banach-Mazur metric: � � � � T : X → Y linear isomorphism � T � · � T − 1 � d ([ X ] , [ Y ]) = ln inf � � dim X = n } , BM ( n ) = { [ X ] the Banach-Mazur compactum. BM 0 ( n ) = BM ( n ) \ { [ E ] } , the punctured Banach-Mazur compactum. HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Theorem (Ant., 2000, Fund. Math.) Let L ( n ) = { A ∈ J ( n ) | A = − A } . Then BM ( n ) ∼ = L ( n ) / O ( n ) . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Theorem (Ant., 2005, Fundamentalnaya i Prikladnaya Matematika) Let the orthogonal group O ( n ) act on a Hilbert cube Q in such a way that: (a) Q is an O ( n ) -AR with a unique O ( n ) -fixed point ∗ , (b) Q is strictly O ( n ) -contractible to ∗ , (c) for a closed subgroup H ⊂ O ( n ) , Q H = {∗} if and only if H acts transitively on the unit sphere S n − 1 and, Q H is homeomorphic to the Hilbert cube whenever Q H � = {∗} , (d) for any closed subgroup H ⊂ O ( n ) , the H-orbit space Q 0 / H is a Q-manifold, where Q 0 = X \ {∗} . Then for every K < O ( n ) , Q 0 / K ∼ = L 0 ( n ) / K. In particular, Q 0 / O ( n ) ∼ = BM 0 ( n ) , and hence, Q / O ( n ) ∼ = BM ( n ) . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva A G -space X is called strictly G -contractible, if there exist a G -homotopy f t : X → X , t ∈ 0 , 1 and a G -fixed point a ∈ X such that f 0 is the identity map of X , and f t ( x ) = a if and only if ( x , t ) ∈ { ( x , 1) , ( a , t ) } . The corresponding nonequivariant notion was introduced by Michael. Corollary J ( n ) / O ( n ) is homeomorphic to the Banach-Mazur compactum BM ( n ) . cb ( R n ) / Aff ( n ) ∼ = J ( n ) / O ( n ) ∼ = BM ( n ) . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva A G -space X is called strictly G -contractible, if there exist a G -homotopy f t : X → X , t ∈ 0 , 1 and a G -fixed point a ∈ X such that f 0 is the identity map of X , and f t ( x ) = a if and only if ( x , t ) ∈ { ( x , 1) , ( a , t ) } . The corresponding nonequivariant notion was introduced by Michael. Corollary J ( n ) / O ( n ) is homeomorphic to the Banach-Mazur compactum BM ( n ) . cb ( R n ) / Aff ( n ) ∼ = J ( n ) / O ( n ) ∼ = BM ( n ) . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva A G -space X is called strictly G -contractible, if there exist a G -homotopy f t : X → X , t ∈ 0 , 1 and a G -fixed point a ∈ X such that f 0 is the identity map of X , and f t ( x ) = a if and only if ( x , t ) ∈ { ( x , 1) , ( a , t ) } . The corresponding nonequivariant notion was introduced by Michael. Corollary J ( n ) / O ( n ) is homeomorphic to the Banach-Mazur compactum BM ( n ) . cb ( R n ) / Aff ( n ) ∼ = J ( n ) / O ( n ) ∼ = BM ( n ) . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva A G -space X is called strictly G -contractible, if there exist a G -homotopy f t : X → X , t ∈ 0 , 1 and a G -fixed point a ∈ X such that f 0 is the identity map of X , and f t ( x ) = a if and only if ( x , t ) ∈ { ( x , 1) , ( a , t ) } . The corresponding nonequivariant notion was introduced by Michael. Corollary J ( n ) / O ( n ) is homeomorphic to the Banach-Mazur compactum BM ( n ) . cb ( R n ) / Aff ( n ) ∼ = J ( n ) / O ( n ) ∼ = BM ( n ) . HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Special Case of exp S 1 Denote exp 0 S 1 = ( exp S 1 ) \ { S 1 } . Corollary (Ant., 2007, Topology Appl.) ( exp 0 S 1 ) / S 1 ∼ = L 0 (2) / S 1 . Corollary (Toru´ nczyk-West, 1978) ( exp 0 S 1 ) / S 1 is a Q-manifold Eilenberg-MacLane space K ( Q , 2) . Proof [Ant., 2007, Topology Appl.] Since ( exp 0 S 1 ) / S 1 ∼ = L 0 (2) / S 1 and L 0 (2) / S 1 is a Q -manifold Eilenberg-MacLane space K ( Q , 2) (Ant., 2000, Fund. Math.) HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Special Case of exp S 1 Denote exp 0 S 1 = ( exp S 1 ) \ { S 1 } . Corollary (Ant., 2007, Topology Appl.) ( exp 0 S 1 ) / S 1 ∼ = L 0 (2) / S 1 . Corollary (Toru´ nczyk-West, 1978) ( exp 0 S 1 ) / S 1 is a Q-manifold Eilenberg-MacLane space K ( Q , 2) . Proof [Ant., 2007, Topology Appl.] Since ( exp 0 S 1 ) / S 1 ∼ = L 0 (2) / S 1 and L 0 (2) / S 1 is a Q -manifold Eilenberg-MacLane space K ( Q , 2) (Ant., 2000, Fund. Math.) HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Special Case of exp S 1 Denote exp 0 S 1 = ( exp S 1 ) \ { S 1 } . Corollary (Ant., 2007, Topology Appl.) ( exp 0 S 1 ) / S 1 ∼ = L 0 (2) / S 1 . Corollary (Toru´ nczyk-West, 1978) ( exp 0 S 1 ) / S 1 is a Q-manifold Eilenberg-MacLane space K ( Q , 2) . Proof [Ant., 2007, Topology Appl.] Since ( exp 0 S 1 ) / S 1 ∼ = L 0 (2) / S 1 and L 0 (2) / S 1 is a Q -manifold Eilenberg-MacLane space K ( Q , 2) (Ant., 2000, Fund. Math.) HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Special Case of exp S 1 Denote exp 0 S 1 = ( exp S 1 ) \ { S 1 } . Corollary (Ant., 2007, Topology Appl.) ( exp 0 S 1 ) / S 1 ∼ = L 0 (2) / S 1 . Corollary (Toru´ nczyk-West, 1978) ( exp 0 S 1 ) / S 1 is a Q-manifold Eilenberg-MacLane space K ( Q , 2) . Proof [Ant., 2007, Topology Appl.] Since ( exp 0 S 1 ) / S 1 ∼ = L 0 (2) / S 1 and L 0 (2) / S 1 is a Q -manifold Eilenberg-MacLane space K ( Q , 2) (Ant., 2000, Fund. Math.) HYPERSPACES OF COMPACT CONVEX SETS

Affine group action on cb ( R n ) Motivation Global Slices The John ellipsoid Computing J ( n ) The Banach-Mazur compacta Equiva Corollary (Ant., 2007, Topology Appl.) ( exp 0 S 1 ) / O (2) ∼ = L 0 (2) / O (2) . ( exp S 1 ) / O (2) ∼ = BM (2) . HYPERSPACES OF COMPACT CONVEX SETS