Abstract

We establish a general approximations theorem for semiconvex and semiconcave functions with general modulus by using the compensated convex transforms introduced by K. Zhang [Compensated convexity and its applications, Ann. Inst. H. Poincaré (C) Non Linear Analysis 25/4 (2008) 743–771]. For a semiconvex function ff 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 [Über die kleinste Kugel, die eine räumliche Figur einschlie{ß}t, J. Reine Angew. Mathematik 123 (1901) 241–257] of the Fréchet sub\-differential. We also prove a C1,ωλC^{1,\omega _\lambda} regularity result for the upper transform of semiconvex functions with general modulus.

Author information

Contact details are reproduced from the original publication and may be historical.

Maryam Salam Alatawi

Department of Mathematics, University of Tabuk, Tabuk 71491, Saudi Arabia

msoalatawi@ut.edu.sa

Keywords

  • Compensated convex transforms
  • convex function
  • semiconvex function
  • semiconcave function
  • linear modulus
  • general modulus
  • singularity extraction
  • minimal bounding sphere
  • Fréchet subdifferential

Mathematics Subject Classification

52A41, 41A30, 26B25, 49J52

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Suggested citation

M. S. Alatawi. “The Regularity Property of Compensated Convex Transforms for Semiconvex Functions of General Modulus.” Journal of Convex Analysis 31 (2024), No. 3, 959–981. https://doi.org/10.68381/jca31046

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