The Core of a Set-Valued Mapping and the Finiteness Principle for - PowerPoint PPT Presentation

6. λ -Balanced Refinements Let F : M → Y be a set-valued mapping, an let λ ≥ 0 . Let � , � BR [ F : λ ]( x ) = � F ( z ) + λ ρ ( x , z ) B Y x ∈ M . z ∈M We refer to the set-valued mapping BR [ F : λ ] : M → K m ( Y ) as a λ -balanced refinement of the mapping F . P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 24 / 112

Clearly, BR [ F : λ ]( x ) is a convex compact subset of Y , and BR [ F : λ ]( x ) ⊂ F ( x ) for all x ∈ M . Let � λ = { λ 0 , λ 1 , ..., λ ℓ } where 1 ≤ λ k ≤ λ k + 1 , k = 1 , ..., ℓ − 1 . We set F [0] = F , and F [ k + 1] ( x ) = BR [ F [ k ] : λ k ]( x ) = � � F [ k ] ( z ) + λ k ρ ( x , z ) B Y � z ∈M for every x ∈ M and k ∈ N . P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 25 / 112

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Conjecture 5. Let m ∈ N . There exist constants ℓ = ℓ ( m ) ∈ N , γ = γ ( m ) ≥ 1 , and a non-decreasing positive sequence of parameters � λ = { λ 0 ( m ) , λ 2 ( m ) , ..., λ ℓ ( m ) } , such that the following holds: Let F : M → K m ( Y ) be a set-valued mapping such that for every subset M ′ ⊂ M with # M ′ ≤ N ( m , Y ) , the restriction F | M ′ of F to M ′ has a Lipschitz selection f M ′ : M ′ → Y with � f M ′ � Lip( M ′ , Y ) ≤ 1 . Then the set-valued mapping F [ ℓ ] : M → K m ( Y ) is a γ -core of F . P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 30 / 112

Recall that F [ ℓ ] is a γ -core if d H ( F [ ℓ ] ( x ) , F [ ℓ ] ( y )) ≤ γ ρ ( x , y ) , x , y ∈ M . Thus, F [ ℓ ] ( x ) ⊂ F [ ℓ ] ( y ) + γ ρ ( x , y ) B Y , x , y ∈ M . Let us reformulate this property in terms of γ -balanced refinements. Given x ∈ M we have: F [ ℓ + 1] ( x ) = BR [ F [ ℓ ] : γ ]( x ) = � � � F [ ℓ ] ( y ) + γ ρ ( x , y ) B Y y ∈M so that F [ ℓ + 1] ( x ) ⊃ F [ ℓ ] ( x ) proving that F [ ℓ + 1] = F [ ℓ ] on M . P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 31 / 112

Conjecture 5.1: Stabilization Property of λ -Balanced Refinements Given m ∈ N there exist ℓ = ℓ ( m ) ∈ N and a non-decreasing positive sequence � λ = { λ 0 ( m ) , λ 2 ( m ) , ..., λ ℓ ( m ) } such that for every set-valued mapping F : M → K m ( Y ) satisfying the hypothesis of the Finiteness Principle the following Stabilization Property F [ ℓ + 1] ( x ) = F [ ℓ ] ( x ) � ∅ x ∈ M , for all holds. P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 32 / 112

Theorem 6. Let ( M , ρ ) be a pseudometric space. Conjecture 5 holds with � λ = { 2 6 , 2 7 } γ = 2 14 ℓ = 2 (two iterations), and whenever: (i) m = 1 and Y is an arbitrary Banach space; (ii) m = 2 and dim Y = 2 . P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 33 / 112

Conjecture 5: m = 2 and dim Y = 2 A Sketch of the Proof. The finiteness constant N (2 , Y ) = 4 provided dim Y = 2 . We know that for every subset M ′ ⊂ M with # M ′ ≤ 4 , the restriction F | M ′ of F to M ′ has a Lipschitz selection f M ′ : M ′ → Y with � f M ′ � Lip( M ′ , Y ) ≤ 1 . Proposition 7. (Sh. [2002]) For every subset S ⊂ M with # S ≤ 10 the restriction F | S of F to S has a Lipschitz selection f S : S → R 2 with the Lipschitz seminorm � f S � Lip( S , R 2 ) ≤ 2 6 . P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 34 / 112

Let B = B Y . We introduce a new metric on M : d( x , y ) = 2 6 ρ ( x , y ) , x , y ∈ M . Then the following assumption holds: Assumption 8. For every subset S ⊂ M with # S ≤ 10 the restriction F | S has a Lipschitz (with respect to d ) selection f S : S → R 2 with the Lipschitz seminorm � f S � Lip(( S , d ) , R 2 ) ≤ 1 . We proceed two balanced refinements of F (with respect to the metric d ) with the parameters � λ = { 1 , 2 } : � F [1] ( x ) = [ F ( z ) + d( x , z ) B ] , x ∈ M , z ∈M and G ( x ) = F [2] ( x ) = BR [ F [1] :2] = � � � F [1] ( z ) + 2 d( x , z ) B , x ∈ M . z ∈M P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 35 / 112

Thus,     �   �   � F ( z ′ ) + d( z , z ′ ) B �     G ( x ) =  + 2 d( x , z ) B  , x ∈ M .                 z ′ ∈M z ∈M Clearly, G ( x ) ⊂ F ( x ) , x ∈ M . We prove that the set-valued mapping G : M → K 2 ( Y ) is a γ − core of F (with respect to d ) with γ = 162 = 2 · 9 2 . Thus, our aim is prove that (i) G ( x ) � ∅ for every x ∈ M ; (ii) d H ( G ( x ) , G ( y )) ≤ γ d( x , y ) for all x , y ∈ M . P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 36 / 112

The proof of part (i) relies on the following corollary of Helly’s Theorem: Lemma 9. Let K be a collection of convex compact subsets of R 2 . Suppose that � K � ∅ . K ∈ K Then for every r ≥ 0 the following equality    �  � � � � K ′ � �    + B (0 , r ) = + B (0 , r ) K K          K , K ′ ∈ K K ∈ K holds. P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 37 / 112

We recall that       �  �  � F ( z ′ ) + d( z , z ′ ) B �      + 2 d( x , z ) B  , x ∈ M . G ( x ) =                 z ′ ∈M z ∈M This and Lemma 9 imply the following representation of the set G ( x ) : Lemma 10. For every x ∈ M � � �� [ F ( z 1 ) + d( z 1 , z ) B ] � � [ F ( z 2 ) + d( z 2 , z ) B ] + 2d( z , x ) B G ( x ) = z , z 1 , z 2 ∈M P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 38 / 112

Given x , z , z 1 , z 2 ∈ M , let � � [ F ( z 1 ) + d( z 1 , z ) B ] � H ( z 1 , z 2 , z : x ) = [ F ( z 2 ) + d( z 2 , z ) B ] + 2 d( z , x ) B . a ∈ H ( z 1 , z 2 , z : x ) ⇐⇒ ∃ g ( z 1 ) ∈ F ( z 1 ) , g ( z 2 ) ∈ F ( z 2 ) , g ( z ) ∈ R 2 , g ( x ) = a , � g ( z ) − g ( z 1 ) � ≤ d( z , z 1 ) , � g ( z ) − g ( z 2 ) � ≤ d( z , z 2 ) , � g ( x ) − g ( z ) � ≤ 2 d( z , x ) . P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 39 / 112

Thus, � H ( z 1 , z 2 , z : x ) G ( x ) = z , z 1 , z 2 ∈M This representation, Helly’s Theorem in R 2 and Assumption 8 readily imply the required property (i): G ( x ) � ∅ , x ∈ M . Prove property (ii) which is equivalent to the following imbeddings: G ( x ) + γ d( x , y ) B ⊃ G ( y ) x , y ∈ M , and G ( y ) + γ d( x , y ) B ⊃ G ( x ) , x , y ∈ M . Given x , y ∈ M let us prove that G ( x ) + γ d( x , y ) B ⊃ G ( y ) P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 40 / 112

Lemma 9 and 10 tell us:     �   G ( x ) + γ d( x , y ) B = H ( z 1 , z 2 , z : x )  + γ d( x , y ) B =            z , z 1 , z 2 ∈M � � � � � � H ( u 1 , u 2 , u : x ) H ( v 1 , v 2 , v : x ) + γ d( x , y ) B A⊂M where A = { u , u 1 , u 2 , v , v 1 , v 2 , x } runs over all subsets of M with # A ≤ 7 . Fix A = { u , u 1 , u 2 , v , v 1 , v 2 , x } ⊂ M . Let � � � S = H ( u 1 , u 2 , u : x ) H ( v 1 , v 2 , v : x ) + γ d( x , y ) B . Prove that � S ⊃ G ( y ) = H ( z 1 , z 2 , z : y ) . z , z 1 , z 2 ∈M P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 41 / 112

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We recall the structure of the set H ( z 1 , z 2 , z : y ) : P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 44 / 112

The proof relies on the following two auxiliary results. Proposition 11. Let C ⊂ Y be a convex set. Let a ∈ Y and let r > 0 . Suppose C ∩ B ( a , r ) � ∅ . Then for every s > 0 C ∩ B ( a , 2 r ) + 9 s B ⊃ ( C + sB ) ∩ ( B ( a , 2 r ) + sB ) . The next pictures illustrate the geometrical background of this imbedding. P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 45 / 112

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Proposition 11 and Helly’s Theorem in R 2 imply the following result. Proposition 12. Let C , C 1 , C 2 ⊂ R 2 be convex subsets, and let r > 0 . Let us assume that C 1 ∩ C 2 ∩ ( C + rB ) � ∅ . Then for every δ > 0 { ( C 1 ∩ C 2 ) + 2 rB } ∩ C + 18 δ B ⊃ [( C 1 ∩ C 2 ) + 2( r + δ ) B ] ∩ [(( C 1 + rB ) ∩ C ) + 2 δ B ] ∩ [(( C 2 + rB ) ∩ C ) + 2 δ B ] P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 59 / 112

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A Sketch of the Proof. Let a ∈ [ C 1 ∩ C 2 + 2( r + δ ) B ] ∩ [( C 1 + rB ) ∩ C + 2 δ B ] ∩ [( C 2 + rB ) ∩ C + 2 δ B ] . Using Helly’s Theorem and the hypothesis of the proposition we prove that there exists a point x ∈ R 2 such that x ∈ C 1 ∩ C 2 ∩ ( C + rB ) ∩ B ( a , 2 r + 2 δ ) . Hence, x ∈ C + rB so that B ( x , r ) ∩ C � ∅ . Proposition 12 tells us that in this case C ∩ B ( x , 2 r ) + 18 δ B ⊃ [ C + 2 δ B ] ∩ [ B ( x , 2 r ) + 2 δ B ] [ C + 2 δ B ] ∩ B ( x , 2 r + 2 δ ) . = P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 61 / 112

Recall that a ∈ [ C 1 ∩ C 2 + 2( r + δ ) B ] ∩ [( C 1 + rB ) ∩ C + 2 δ B ] ∩ [( C 2 + rB ) ∩ C + 2 δ B ] , x ∈ C 1 ∩ C 2 ∩ ( C + rB ) ∩ B ( a , 2 r + 2 δ ) . Then x ∈ B ( a , 2 r + 2 δ ) so that a ∈ B ( x , 2 r + 2 δ ) . Furthermore, a ∈ [( C 1 + rB ) ∩ C ] + 2 δ B ⊂ C + 2 δ B = ⇒ ( C + 2 δ B ) ∩ B ( x , 2 r + 2 δ ) ∋ a . Hence, C ∩ B ( x , 2 r ) + 18 δ B ⊃ [ C + 2 δ B ] ∩ B ( x , 2 r + 2 δ ) ∋ a . But x ∈ C 1 ∩ C 2 which proves the required inclusion [( C 1 ∩ C 2 ) + 2 rB ] ∩ C + 18 δ B ∋ a . � P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 62 / 112

We return to the proof of the imbedding � � � � H ( u 1 , u 2 , u : x ) H ( v 1 , v 2 , v : x ) + γ d( x , y ) B ⊃ H ( z 1 , z 2 , z : y ) . S = z , z 1 , z 2 ∈M We recall that � � [ F ( u 1 ) + d( u 1 , z ) B ] � [ F ( u 2 ) + d( u 2 , z ) B ] H ( u 1 , u 2 , u : x ) = + 2 d( u , x ) B and � � [ F ( v 1 ) + d( v 1 , v ) B ] � [ F ( v 2 ) + d( v 2 , v ) B ] H ( v 1 , v 2 , v : x ) = + 2 d( v , x ) B . P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 63 / 112

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To apply Proposition 12 to the set S we have to check that C 1 ∩ C 2 ∩ ( C + rB ) � ∅ . We know that the restriction F | B of F to the set B = { u 1 , u 2 , u , v 1 , v 2 , v , x , } has a Lipschitz selection f : B → R 2 with � f � Lip( B , R 2 ) ≤ 1 . Then, C 1 ∩ C 2 ∩ ( C + rB ) ∋ f ( u ) proving that the hypothesis of Proposition 12 holds. P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 69 / 112

By this proposition, S = ( C 1 ∩ C 2 + 2 rB ) ∩ C + 18 δ B ⊃ [( C 1 ∩ C 2 ) + 2( r + δ ) B ] ∩ [(( C 1 + rB ) ∩ C ) + 2 δ B ] ∩ [(( C 2 + rB ) ∩ C ) + 2 δ B ] = A 1 ∩ A 2 ∩ A 3 . Prove that A 1 = ( C 1 ∩ C 2 ) + 2( r + δ ) B ⊃ G ( y ) , A 2 = (( C 1 + rB ) ∩ C ) + 2 δ B ⊃ G ( y ) , and A 3 = (( C 2 + rB ) ∩ C ) + 2 δ B ⊃ G ( y ) . P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 70 / 112

Prove that A 1 = ( C 1 ∩ C 2 ) + 2( r + δ ) B ⊃ H ( u 1 , u 2 , u : y ) . Recall that A 1 = ( C 1 ∩ C 2 ) + 2( r + δ ) B = { F ( u 1 ) + d( u 1 , u ) B } ∩ { F ( u 2 ) + d( u 2 , u ) B } + 2(d( u , x ) + 9 d( x , y )) B . By the triangle inequality, d( u , x ) + 9 d( x , y ) ≥ d( u , x ) + d( x , y ) ≥ d( u , y ) so that A 1 = ( C 1 ∩ C 2 ) + 2( r + δ ) B ⊃ { F ( u 1 ) + d( u 1 , u ) B } ∩ { F ( u 2 ) + d( u 2 , u ) B } + 2 d( u , y ) B = H ( u 1 , u 2 , u : y ) ⊃ G ( y ) . P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 71 / 112

Prove that A 2 = (( C 1 + rB ) ∩ C ) + 2 δ B ⊃ G ( y ) . P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 72 / 112

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Applying Proposition 12 we obtain the required inclusion A 2 ⊃ H ( v 1 , v 2 , v : y ) ∩ H ( u 1 , v 1 , x : y ) ∩ H ( u 1 , v 2 , x : y ) ⊃ G ( y ) . In the same fashion we show that A 3 = [(( C 2 + rB ) ∩ C ) + 2 δ B ] ⊃ G ( y ) proving the required imbedding G ( x ) + γ d( x , y ) B ⊃ G ( y ) with γ = 2 · 9 2 = 162 . By interchanging the roles of x and y we obtain also G ( y ) + γ d( x , y ) B ⊃ G ( x ) . Hence, d H ( G ( x ) , G ( y )) ≤ γ d( x , y ) = 2 6 γρ ( x , y ) , x , y ∈ M , proving that the set-valued mapping G is a 2 6 γ -core of F . � P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 84 / 112

7. Lipschitz Selection in R 2 : an Algorithm. The proof of Theorem 6 provides an efficient algorithm for constructing of an almost optimal Lipschitz selection for any set-valued mapping F : M → K 2 ( R 2 ) satisfying the hypothesis of the Finiteness Principle. • Y = ℓ 2 ∞ = ( R 2 , � · � ) , where � x � = max {| x 1 | , | x 2 |} for x = ( x 1 , x 2 ) ∈ R 2 ; • Q 0 = [ − 1 , 1] × [ − 1 , 1] ; • “box” or “rectangle” - a rectangle in R 2 with sides parallel to the coordinate axes; • R ( R 2 ) - the family of all “boxes” in R 2 . • Given G ⊂ R 2 we let H [ G ] denote the smallest box containing G : Π = [ a , b ] × [ c , d ] ⊂ R 2 : Π ⊃ G � � � H [ G ] = P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 85 / 112

Let ( M , ρ ) be a pseudometric space, and let F : M → K 2 ( R 2 ) be a set-valued mapping satisfying the following condition: There exists a constant α > 0 such that for every subset M ′ ⊂ M with # M ′ ≤ 4 the restriction F | M ′ has a Lipschitz selection f M ′ : M ′ → R 2 with the Lipschitz seminorm � f S � Lip( M ′ , R 2 ) ≤ α. STEP 1. We construct a 2 6 α -balanced refinement of F : � � � F [1] ( x ) = F ( y ) + 2 6 α ρ ( x , y ) Q 0 , x ∈ M . y ∈M STEP 2. We construct a 2 7 α -balanced refinement of F [1] : � � � F [2] ( x ) = F [1] ( y ) + 2 7 α ρ ( x , y ) Q 0 , x ∈ M . y ∈M P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 86 / 112

STEP 3. We construct a set-valued mapping H F : M → R ( R 2 ) which to every x ∈ M assigns the smallest box containing F [2] ( x ) : � � F [2] ( x ) H F ( x ) = H , x ∈ M . STEP 4. We define a Lipschitz selection f : M → R 2 of F by � � � � F [2] ( x ) f ( x ) = center ( H F ( x )) = center , x ∈ M . H Here given a rectangle P ∈ R ( R 2 ) we let center ( P ) denote the center of P . P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 87 / 112

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The following statement justifies STEP 3 and STEP 4 of the Algorithm. Statement 14. Let G ⊂ R 2 be a convex compact set. (1) Then center ( H ( G )) ∈ G . Let G 1 , G 2 ⊂ R 2 be convex compact sets. Then (2) d H ( H [ G 1 ] , H [ G 2 ]) ≤ d H ( G 1 , G 2 ) . (3) For every two boxes P 1 , P 2 ∈ R ( R 2 ) we have � center ( P 1 ) − center ( P 2 ) � ≤ d H ( P 1 , P 2 ) . (Recall that R 2 is equipped with the ℓ 2 ∞ -norm.) We know that the set-valued mapping F [2] : M → K 2 is a γ -core of F with γ = 2 14 α , i.e., d H ( F [2] ( x ) , F [2] ( y )) ≤ γ ρ ( x , y ) , x , y ∈ M . Combining this inequality with Statement 14 we conclude that f is a Lipschitz selection of F with � f � Lip( M , R 2 ) ≤ γ. P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 93 / 112

8. Criterions for Lipschitz Selections in R 2 Let Y = ℓ 2 ∞ , and let F : M → K ( R 2 ) be a set valued mapping. Given λ > 0 and x , x ′ ∈ M , let R λ [ x , x ′ : F ] = H [ F ( x ) ∩ { F ( x ′ ) + λ ρ ( x , x ′ ) Q 0 } ] . P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 94 / 112

8. Criterions for Lipschitz Selections in R 2 Let Y = ℓ 2 ∞ , and let F : M → K ( R 2 ) be a set valued mapping. Given λ > 0 and x , x ′ ∈ M , let R λ [ x , x ′ : F ] = H [ F ( x ) ∩ { F ( x ′ ) + λ ρ ( x , x ′ ) Q 0 } ] . P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 95 / 112

8. Criterions for Lipschitz Selections in R 2 Theorem 15 (Sh. [2002]) A set-valued mapping F : M → K ( R 2 ) has a Lipschitz selection if and only if ∃ λ > 0 such that: (i) R λ [ x , x ′ : F ] � ∅ x , x ′ ∈ M ; for every (ii) For every x , x ′ , y , y ′ ∈ M the following inequality dist � R λ [ x , x ′ : F ] , R λ [ y , y ′ : F ] � ≤ λ ρ ( x , y ) holds. Furthermore, inf {� f � Lip( M , R 2 ) : f is a selection of F on M} ∼ inf λ P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 96 / 112

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P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 98 / 112

P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 99 / 112

This criterion follows from a proof of the Finiteness Principle for Lipschitz selections for Y = R 2 given below. Given a set-valued mapping F : M → K 2 ( R 2 ) , we assume that the restriction F | M ′ of F to every M ′ ⊂ M with # M ≤ 4 has a Lipschitz selection f M ′ : M ′ → R 2 with � f M ′ � Lip( M ′ , R 2 ) ≤ 1 . Prove that F has a Lipschitz selection f : M → R 2 with � f � Lip( M , R 2 ) ≤ 8 . A Sketch of the Proof. STEP 1. We construct the 1 -balanced refinement of the mapping F : � F ( y ) + ρ ( x , y ) B � , � F [1] ( x ) = x ∈ M . y ∈M STEP 2. We define a set-valued mapping T F : M → R ( R 2 ) which to every x ∈ M assigns the smallest box containing F [1] ( x ) : � F [1] ( x ) � T F ( x ) = H , x ∈ M . P . Shvartsman (Technion, Haifa, Israel) The core of a set-valued mapping August 5-9, 2019 100 / 112