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The quasiconvex envelope through first-order partial differential equations which characterize quasiconvexity of nonsmooth functions

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  • Necessary and sufficient conditions for quasiconvexity, also called level-set convexity, of a function are given in terms of first-order partial differential equations. Solutions to the equations are understood in the viscosity sense and the conditions apply to nonsmooth and semicontinuous functions. A comparison principle, implying uniqueness of solutions, is shown for a related partial differential equation. This equation is then used in an iterative construction of the quasiconvex envelope of a function. The results are then extended to robustly quasiconvex functions, that is, functions which are quasiconvex under small linear perturbations.
    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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