Extreme Points of Some Convex Subsets of L 1 (0, 1)

Ryff, John V.

doi:10.2307/2035788

Cited by 20 publications

(32 citation statements)

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“…The next statement is originally due to Ryff [18] in the case (Ω, F, P) is [0, 1] endowed with its Borel field and the Lebesgue measure, we believe it is well-known but give a short proof for the sake of completeness: Proof. Assume first that Z n is of the form mentioned above and Z n converges a.s. to X, then for each concave u, one has E(u(Z n )) ≥ E(u(Y )) and one concludes with Lebesgue's dominated convergence theorem (the Z n 's being uniformly bounded).…”

Section: Preliminariesmentioning

confidence: 93%

A numerical approach for a class of risk-sharing problems

Carlier¹,

Lachapelle²

2011

Journal of Mathematical Economics

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Section: Preliminariesmentioning

confidence: 93%

A numerical approach for a class of risk-sharing problems

Carlier¹,

Lachapelle²

2011

Journal of Mathematical Economics

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“…The idea of rearranging a function dates back to the book [19] of Hardy, Littlewood and Pólya, since than many authors have investigated both extensions and applications of this notion. Here we relies on the results in [1,7,8,14,22,26].…”

Section: Rearrangements Of Measurable Functionsmentioning

confidence: 99%

Steiner symmetry in the minimization of the first eigenvalue of a fractional eigenvalue problem with indefinite weight

Anedda

Cuccu

Frassu

2020

Can. J. Math.-J. Can. Math.

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Let Ω ⊂ R N , N ≥ 2, be an open bounded connected set. We consider the fractional weighted eigenvalue problem (−∆) s u = λρu in Ω with homogeneous Dirichlet boundary condition, where (−∆) s , s ∈ (0, 1), is the fractional Laplacian operator, λ ∈ R and ρ ∈ L ∞ (Ω). We study weak* continuity, convexity and Gâteaux differentiability of the map ρ → 1/λ 1 (ρ), where λ 1 (ρ) is the first positive eigenvalue. Moreover, denoting by G(ρ 0 ) the class of rearrangements of ρ 0 , we prove the existence of a minimizer of λ 1 (ρ) when ρ varies on G(ρ 0 ). Finally, we show that, if Ω is Steiner symmetric, then every minimizer shares the same symmetry.Recently, great attention has been focused on the study of fractional and nonlocal operators of elliptic type, both for pure mathematical research and in view of concrete real-world applications. This type of operators arises in a quite natural way in many different contexts, such as, among others, the thin obstacle problem,

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“…Here g is the corresponding Lie algebra of smooth functions with the Poisson bracket and t ⊆ g is the abelian subalgebra of radial functions f (r, θ) = f (r). In this context several completion procedures are required to identify the natural generalization of the Weyl group, which in this context leads to the semigroup W of measure preserving maps on [0, 1] and the closed convex hulls of its orbits in L 1 ([0, 1]) ( [Br66], [Ry65,Ry67]); see also Problem 5.7). This exhibits an interesting analogy with the situation discussed in Remark 5.6 below, where we show that the monoid Isom(H) acts on orbit closures in the duals spaces u p (H) ′ and u(H) ′ .…”

Section: Neebmentioning

confidence: 99%

Semibounded Representations and Invariant Cones in Infinite Dimensional Lie Algebras

Neeb

2010

Confluentes Math.

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For the Lie algebra g of a connected infinite-dimensional Lie group G, there is a natural duality between so-called semi-equicontinuous weak- * -closed convex Ad * (G)-invariant subsets of the dual space g ′ and Ad(G)-invariant lower semicontinuous positively homogeneous convex functions on open convex cones in g. In this survey, we discuss various aspects of this duality and some of its applications to a more systematic understanding of open invariant cones and convexity properties of coadjoint orbits. In particular, we show that root decompositions with respect to elliptic Cartan subalgebras provide powerful tools for important classes of infinite Lie algebras, such as completions of locally finite Lie algebras, Kac-Moody algebras and twisted loop algebras with infinite-dimensional range spaces. We also formulate various open problems. Mathematics Subject Classification: 22E65, 22E45

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Extreme Points of Some Convex Subsets of L 1 (0, 1)

Cited by 20 publications

References 0 publications

A numerical approach for a class of risk-sharing problems

A numerical approach for a class of risk-sharing problems

Steiner symmetry in the minimization of the first eigenvalue of a fractional eigenvalue problem with indefinite weight

Semibounded Representations and Invariant Cones in Infinite Dimensional Lie Algebras

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