A new proof of the integrability   of the subdifferential   of
a convex function on a Banach space

Ivanov, Milen; Zlateva, Nadia

doi:10.1090/s0002-9939-08-09178-8

Cited by 2 publications

(2 citation statements)

References 13 publications

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“…In the Euclidean framework, as well as in the more general Banach setting, a convex function can be detected using its subdifferential at every point via the Rockafellar function (for a new and recent proof, see [16]). We are going to prove that a similar integrability property is inherited by convex functions on the Heisenberg group, where the H-subdifferential plays nearly the same role.…”

Section: The Rockafellar Function In Ihmentioning

confidence: 99%

Horizontal normal map on the Heisenberg group

Calogero,

Pini

2008

Preprint

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We investigate the notion of H-subdifferential and H-normal map of a function u on the Heisenberg group, based on the sub-Riemannian structure of IH. We show that some properties of the subdifferential in the Euclidean setting are inherited. In particular, a characterization of the convexity of a function is given via the nonemptiness of the H-subdifferential ∂ H u(g) at every point g. Concerning the H-normal map, we prove a monotonicity result when suitable strictly convex radial functions are considered. Finally, we suggest a definition of the Monge-Ampère measure of a function u via its H-normal map, and we extend a well-known integration result by Rockafellar.

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Section: The Rockafellar Function In Ihmentioning

confidence: 99%

Horizontal normal map on the Heisenberg group

Calogero,

Pini

2008

Preprint

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“…Since then, this property has been considered for various classes of functions; let us mention the works of Correa-Jofré [9], Qi [23,24], Birge-Qi [3], Thibault-Zagrodny [30,31,32], Borwein-Moors [7], Thibault-Zlateva [33], Bernard-Thibault-Zagrodny [2], Zajíček [34] and our recent work [17]. The subdifferential representation of extended-real-valued lower semicontinuous convex functions defined on a Banach space was established by Rockafellar [25]; different proofs of this result are proposed by Taylor [28], Thibault [29] and Ivanov-Zlateva [13], and a refined version by Benoist-Daniilidis [1]. Few results exist for non-convex functions; let us mention Qi [23] and Birge-Qi [3].…”

Section: Introductionmentioning

confidence: 99%