The Locality Property for Compensated Convex Transforms

We establish the locality property for compensated convex transforms to functions under various continuity and smoothness. More precisely, we generalise the locality property and the density property for compensated convex transforms obtained recently by K. Zhang, A. Orlando and E. Crooks [Compensated convexity and Hausdorff stable geometric singularity extractions, Math. Models Methods Appl. Sci. 25/4 (2015) 747--801] for bounded functions to functions under various continuity and smoothness assumptions such as Cα, C1,β. We also establish some localised versions of such locality property.