The operation of infimal convolution
Thomas Strömberg · Epubl LTU · 1996
In this well-written paper on infimal convolution the author's purpose is "to provide a survey of the subject as well as to complement or sharpen existing results". The author has chosen a self-contained presentation, where, generally, proofs are given for his original results (spread in several articles). The paper is organized in five sections: 1. Introduction and preliminaries, 2. Elementary properties, 3. The convex case, 4. Continuity of the operation of infimal convolution, 5. Regularization. An important concern is the regularizing effect of the infimal convolution. So, under some additional hypotheses, $f abla g$ is (upper semi-) continuous, uniformly continuous (on bounded sets), Gateaux (Fr\'{e}chet) differentiable, with uniformly (or $p$-H\"{o}lder) continuous derivative if $g$ is so. The continuity properties of the operation of infimal convolution are studied with respect to the epi-convergence, the slice topology, the affine topology and the Attouch-Wets topology. The last section is dedicated to Moreau-Yosida and Lasry-Lions approximations as well as to a generalized Moreau-Yosida approximation. The results established in the previous sections are applied to these particular infimal convolutes, leading to several interesting results.