
Journal of Convex Analysis 26 (2019), No. 1, 275298 Copyright Heldermann Verlag 2019 Fixed Points of LegendreFenchel Type Transforms Alfredo N. Iusem IMPA  Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, 22460320 Rio de Janeiro, Brazil iusp@impa.br Daniel Reem Dept. of Mathematics, The Technion  Israel Institute of Technology, 3200003 Haifa, Israel dream@technion.ac.il Simeon Reich Dept. of Mathematics, The Technion  Israel Institute of Technology, 3200003 Haifa, Israel sreich@technion.ac.il A recent result characterizes the fully order reversing operators acting on the class of lower semicontinuous proper convex functions in a real Banach space as certain linear deformations of the LegendreFenchel transform. Motivated by the Hilbert space version of this result and by the wellknown result saying that this convex conjugation transform has a unique fixed point (namely, the normalized energy function), we investigate the fixed point equation in which the involved operator is fully order reversing and acts on the abovementioned class of functions. It turns out that this nonlinear equation is very sensitive to the involved parameters and can have no solution, a unique solution, or several (possibly infinitely many) ones. Our analysis yields a few byproducts, such as results related to positive definite operators, and to functional equations and inclusions involving monotone operators. Keywords: Convex conjugation, fixed point, functional equation, lower semicontinous proper convex function, LegendreFenchel transform, monotone operator, order reversing operator, positive definite, quadratic function. MSC: 47H10, 26B25, 52A41, 47N10, 47H05, 47J05, 39B42 [ Fulltextpdf (165 KB)] for subscribers only. 