# Curvature of Cost and Production Function

I have been trying to understand the duality theory between cost and production function. Please let me know if my following statement is correct;

If the cost function $C(w,q)$ is convex in $q$, the dual production function $f(x)$ that we can derive will be quasi-concave and it will be unique. However if the cost function is not convex, then the dual production function that we can derive will be quasi-concave but there are more than one such production functions that could have given rise to the cost function.