Let's say I have a convex objective function. The boilerplate example is $z=x^2+y^2$. Now, I also have some constraint, $f(x,y)=0$. Is it true that the constrained optimization problem must be convex as well? I suspect this is the case, but have only rough intuition to back it up and was wondering if there is a proof.
My intuitive argument is that if the objective function is convex, it is always "curved downwards" (where "down" is in the direction of the objective function). Now, since the constraint can only add a condition in the x-y plane, it can't introduce any curvature in the z-direction (the direction of the objective function). So, the convexity of the objective function in that direction must be preserved.