# Compensated convexity and its applications

### Kewei Zhang

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

## Abstract

We introduce the notions of lower and upper quadratic compensated convex transforms $C_{2,λ}(f)$ and $C_{2,λ}(f)$ respectively and the mixed transforms by composition of these transforms for a given function $f:R_{n}↦R$ and for possibly large $λ>0$. We study general properties of such transforms, including the so-called ‘tight’ approximation of $C_{2,λ}(f)$ to *f* as $λ→+∞$ 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 $C_{2,λ}(dist_{2}(x,K))$ of the squared-distance function to a compact set *K* and the quadratic upper transform $C_{2,λ}(f)$ for any convex function *f* of at most quadratic growth. We show that both $C_{2,λ}(dist_{2}(x,K))$ and $C_{2,λ}(f)$ are $C_{1,1}$ approximations of the original functions for large $λ>0$ and $C_{2,λ}(f)$ 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