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
jonathan.borwein@newcastle.edu.au
Liangjin Yao
CARMA, University of Newcastle, Newcastle, NSW 2308, Australia
liangjin.yao@newcastle.edu.au
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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