Conditions for Convexity of Quasiconvex Functions

Necessary and sufficient conditions for convexity of lower semi-continuous quasiconvex functions are given. By applying these results to positively homogeneous functions it is shown that if f is a quadratic form which is quasiconvex on a convex C then f is convexifiable, that is, there exists a strictly increasing function k such that k ∘ f is convex.