Journal of Convex Analysis 31 (2024), No. 3, 959--981
Copyright Heldermann Verlag 2024
The Regularity Property of Compensated Convex Transforms for Semiconvex Functions of General Modulus
Maryam Salam Alatawi
Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia
msoalatawi@ut.edu.sa
[Abstract-pdf]
We establish a general approximations theorem for semiconvex and semiconcave functions with general modulus by using the compensated convex transforms introduced by K.\,Zhang [{\it Compensated convexity and its applications}, Ann. Inst. H. Poincar\'e (C) Non Linear Analysis 25/4 (2008) 743--771]. For a semiconvex function $f$ with general modulus, we show that the limit of the gradient of the upper compensated transform exists and is equal to the center of the minimal bounding sphere in the sense of H.\,Jung [{\it \"Uber die kleinste Kugel, die eine r\"aumliche Figur einschlie{\ss}t}, J. Reine Angew. Mathematik 123 (1901) 241--257] of the Fr\'{e}chet sub\-differential. We also prove a $C^{1,\omega _\lambda}$ regularity result for the upper transform of semiconvex functions with general modulus.
Keywords: Compensated convex transforms, convex function, semiconvex function, semiconcave function, linear modulus, general modulus, singularity extraction, minimal bounding sphere, Frechet subdifferential.
MSC: 52A41, 41A30, 26B25, 49J52.
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