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The Semiconvex Regularization of Functions
A variant of the compensated convexity process introduced and studied by Zhang and his colleagues is considered. It makes amenable the results and tools from convex analysis. It allows the regularization of functions with a strong decay, a new feature. It yields a family of tight approximations of the function that are always semiconvex functions. Thus, these approximating functions have interesting regularity properties. Their convergence is obtained under various assumptions. Additional questions and geometric applications that are important for several fields such as image analysis are deferred to the papers [H. V. Ngai, Proximal Subgradient Methods via the Semiconvex Regularization, in preparation], [H. V. Ngai and J.-P. Penot, Questions About the Semiconvex Regularization Method, in preparation], [H. V. Ngai and J.-P. Penot, Combining Different Regularization Methods, in preparation], and [H. V. Ngai and J.-P. Penot, The Medial Set and the Landscape Function of a Set, in preparation].
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