In the case of real-valued functions of one variable (which is the case at hand), if the function has a minimum and is convex then the minimum is global. If the function is strictly convex (also this case) then it will have at most one global minimum.

True. To keep the explanation simple I glossed over some details. The problem you describe exists even in the 1D case, e.g. the function f(x) = 0. However if a convex function has multiple minima then the minima will be connected and at the same height, so the algorithms still apply.