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Journal of Convex Analysis 21 (2014), No. 1, 261--288
Copyright Heldermann Verlag 2014

Legendre-Type Integrands and Convex Integral Functions

Jonathan M. Borwein
CARMA, University of Newcastle, Newcastle, NSW 2308, Australia

Liangjin Yao
CARMA, University of Newcastle, Newcastle, NSW 2308, Australia

We study the properties of integral functionals induced on L1E (S,μ) by closed convex functions on a Euclidean space E. We give sufficient conditions for such integral functions to be strongly rotund (well-posed). We show that in this generality functions such as the Boltzmann-Shannon entropy and the Fermi-Dirac entropy are strongly rotund. We also study convergence in measure and give various limiting counterexample.

Keywords: Legendre function, monotone operator, set-valued operator, strongly rotund function, Kadec-Klee property, subdifferential operator, Visintin theorem, Vitali's covering theorem, weak convergence, weak compactness, convergence in measure.

MSC: 46B20, 34H05; 47H05, 47N10, 90C25

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