We study a quasiconvex conjugation that transforms the level sum of functions into the pointwise sum of their conjugates and derivenew duality results for the minimization of the max of two quasiconvex functions. Following Barron and al., we show that the levelsum provides quasiconvex viscosity solutions for Hamilton-Jacobi equations in which the initial condition is a general continuousquasiconvex function which is not necessarily Lipschitz or bounded.