# Duality for the level sum of quasiconvex functions and applications

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volle.dvi ESAIM� Control� Optimisation and Calculus of Variations URL� http���www�emath�fr�cocv� September ����� Vol� � �� � DUALITY FOR THE LEVEL SUM OF QUASICONVEX FUNCTIONS AND APPLICATIONS M� VOLLE Abstract� We study a quasiconvex conjugation that transforms the level sum of functions into the pointwise sum of their conjugates and derive new duality results for the minimization of the max of two qua� siconvex functions� Following Barron and al�� we show that the level sum provides quasiconvex viscosity solutions for Hamilton�Jacobi equa� tions in which the initial condition is a general continuous quasiconvex function which is not necessarily Lipschitz or bounded� �� Introduction The powerful properties of the Legendre�Fenchel conjugation for convex functions can be summarized by saying that� under some mild assumptions �also called quali�cation conditions�� it exchanges the pointwise sum of func� tions with the in�mal convolution of their conjugates and vice versa ��� � ��� � The geometrical interpretation of the in�mal convolution of two ex� tended real�valued functions is well known� it lies in the fact that the vecto� rial �or Minkowski� sum of the strict epigraphs of two extended real�valued functions is nothing but the strict epigraph of their in�mal convolution By the way such a property is at the origin of another terminology in which the in�mal convolution is called epigraphical sum ��� �� � When dealing with quasiconvex functions this operation su�ers from the fact that the epi� graphical sum of two quasiconvex functions is no longer quasiconvex ���� � However� there exists an interesting substitute for the epigraphical sum in the �eld of quasiconvexity� namely� the level sum of functions� that is the function whose strict lower level sets are the vectorial sum of the strict lower level sets of the initial functions ���� ��� ��� ��� � � Thus the level sum of two quasiconvex functions is still quasiconvex

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