About the critical points of quasi-convex functions

Question

What do we know about the structure of critical points of quasi-convex functions?

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I am looking for statements like "the critical points of a quasi-convex function are always either a global minima or a saddle point" or "for a quasi-convex function the number of negative eigenvalues of its Hessian at a critical point is constant across intervals of values of the function evaluated at the critical points"

Are statements like this known?

Answer 1

A local minimum of a quasi-convex function is a global minimum. At a critical point where the function is $C^2$, the Hessian matrix is positive semidefinite; if such a critical point is not a local minimum, the Hessian matrix must be singular there. An example of this is the function $f(x) = x^3$.