Legendre-Type Integrands and Convex Integral Functions

Jonathan M. Borwein; Liangjin Yao

Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

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

Mathematics Subject Classification

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

Suggested citation

J. M. Borwein, L. Yao. “Legendre-Type Integrands and Convex Integral Functions.” Journal of Convex Analysis 21 (2014), No. 1, 261–288.