[ubj]

I'm assuming you're referring to nonlinear f(x) because this statement is trivially false for linear f(x).

But consider the function f(x) = max(0, g(x) - c) where the following holds:

* g(x) is nonlinear, positive definite in x, and convex.

* c > 0

Then f(x) is nonlinear and convex (it's the pointwise maximum of convex functions), and the set {x : f(x) = 0} is a convex set.