Existence of solutions and star-shapedness in Minty variational inequalities

Minty variational inequalities have proven to define a stronger notion of equilibrium than Stampacchia variational inequalities. This conclusion leads to argue that some regularity, e.g. convexity or generalized convexity, has to be implicit for any function that admits a solution of the corresponding integrable Minty variational inequality. Quasi-convexity arises almost naturally when functions of one variable are involved. However some differences appear when considering functions of several variables. In this case we show that existence of a solution does not necessarily imply quasi-convexity of the function and instead we prove that the level sets of the function must be star-shaped at a point which is a solution of the Minty variational inequality.