Compensated convexity and its applications

  • Kewei Zhang

    Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, UK


We introduce the notions of lower and upper quadratic compensated convex transforms and respectively and the mixed transforms by composition of these transforms for a given function and for possibly large . We study general properties of such transforms, including the so-called ‘tight’ approximation of to f as and compare our transforms with the well-known Moreau–Yosida regularization (Moreau envelope) and the Lasry–Lions regularization. We also study analytic and geometric properties for both the quadratic lower transform of the squared-distance function to a compact set K and the quadratic upper transform for any convex function f of at most quadratic growth. We show that both and are approximations of the original functions for large and remains convex. Explicitly calculated examples of quadratic transforms are given, including the lower transform of squared distance function to a finite set and upper transform for some non-smooth convex functions in mathematical programming.

Cite this article

Kewei Zhang, Compensated convexity and its applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008), no. 4, pp. 743–771

DOI 10.1016/J.ANIHPC.2007.08.001