Sets Which Split Families of Measurable Sets

Kirk, R. B.

doi:10.1080/00029890.1972.11993146

Cited by 9 publications

(5 citation statements)

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“…x ∈ R d . Then it follows from Proposition 2.6(v) and ( 19), (20) above that ∂ϕ(x) = F ∇ϕ (x) = F f (x) = Φ(x) for all x ∈ R d .…”

Section: Characterization Of Clarke Subdifferentialsmentioning

confidence: 90%

“…Before we proceed, we shall need the following classical result, whose proof is provided for completeness. According to the terminology of Kirk [20], the result asserts the existence, for every Euclidean space, of a countable partition that splits the family of open sets. For alternative proofs, or proofs of similar statements see [25], [12], [11].…”

Section: Cusco Maps and Filippov Representabilitymentioning

confidence: 99%

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Characterization of Filippov representable maps and Clarke subdifferentials

Bivas¹,

Daniilidis²,

Quincampoix³

2020

Preprint

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The ordinary differential equation ẋ(t) = f (x(t)), t ≥ 0, for f measurable, is not sufficiently regular to guarantee existence of solutions. To remedy this we may relax the problem by replacing the function f with its Filippov regularization F f and consider the differential inclusion ẋ(t) ∈ F f (x(t)) which always has a solution. It is interesting to know, inversely, when a setvalued map Φ can be obtained as the Filippov regularization of a (single-valued, measurable) function. In this work we give a full characterization of such set-valued maps, hereby called Filippov representable. This characterization also yields an elegant description of those maps that are Clarke subdifferentials of a Lipschitz function.

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“…x ∈ R d . Then it follows from Proposition 2.6(v) and ( 19), (20) above that ∂ϕ(x) = F ∇ϕ (x) = F f (x) = Φ(x) for all x ∈ R d .…”

Section: Characterization Of Clarke Subdifferentialsmentioning

confidence: 90%

Section: Cusco Maps and Filippov Representabilitymentioning

confidence: 99%

Characterization of Filippov representable maps and Clarke subdifferentials

Bivas¹,

Daniilidis²,

Quincampoix³

2020

Preprint

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show abstract

“…where λ denotes the Lebesgue measure. An explicit construction of such a splitting set can be found in [17] in a general setting (atomless measure space). In the next section we shall enhance this construction to the particular case of a real line and come up with a countable family of disjoint spitting sets.…”

Section: Preliminaries Notationmentioning

confidence: 99%

Linear Structure of Functions with Maximal Clarke Subdifferential

Daniilidis¹,

Flores²

2019

SIAM J. Optim.

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It is hereby established that the set of Lipschitz functions f : U → R (U nonempty open subset of ℓ 1 d ) with maximal Clarke subdifferential contains a linear subspace of uncountable dimension (in particular, an isometric copy of ℓ ∞ (N)). This result goes in the line of a previous result of Borwein-Wang ([8], [9]). However, while the latter was based on Baire category theorem, our current approach is constructive and is not linked to the uniform convergence. In particular we establish lineability (and spaceability for the Lipschitz norm) of the above set inside the set of all Lipschitz continuous functions.

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“…The first definition and construction of a splitting set goes back to [26], while the first examples of Clarke saturated functions can be found in [20,35]. The basic construction proceeds as follows.…”

Section: Definition 1 (Splitting Set) a Measurable Setmentioning

confidence: 99%