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When some variational properties force convexity

M. Volle, J.-B. Hiriart-Urruty, C. Zălinescu (2013)

ESAIM: Control, Optimisation and Calculus of Variations

The notion of adequate (resp. strongly adequate) function has been recently introduced to characterize the essentially strictly convex (resp. essentially firmly subdifferentiable) functions among the weakly lower semicontinuous (resp. lower semicontinuous) ones. In this paper we provide various necessary and sufficient conditions in order that the lower semicontinuous hull of an extended real-valued function on a reflexive Banach space is essentially strictly convex. Some new results on nearest...

Whitney type inequality, pointwise version

Yu. A. Brudnyi, I. E. Gopengauz (2013)

Studia Mathematica

The main result of the paper estimates the asymptotic behavior of local polynomial approximation for L p functions at a point via the behavior of μ-differences, a generalization of the kth difference. The result is applied to prove several new and extend classical results on pointwise differentiability of L p functions including Marcinkiewicz-Zygmund’s and M. Weiss’ theorems. In particular, we present a solution of the problem posed in the 30s by Marcinkiewicz and Zygmund.

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