Non Maximal Cyclically Monotone Graphs and Construction of a Bipotential for the Coulomb's Dry Friction Law

We show a surprising connection between a property of the inf convolution of a family of convex lsc functions and the fact that the intersection of maximal cyclically monotone graphs is the critical set of a bipotential. We then extend our previous results published in this journal [J. Convex Analysis 15(1) (2008) 87–104] to bipotentials convex covers, generalizing the notion of a bi-implicitly convex lagrangian cover. As an application we prove that the bipotential related to Coulomb's friction law is related to a specific bipotential convex cover with the property that any graph of the cover is non maximal cyclically monotone