Convex minimization problem for finding global minimum of a function

Question

I want to minimize $f(x)$ w.r.t to some contraint $h(x)=0$ , here both $f(x)$ and $h(x)$ are convex functions. However, suppose I also know that the function $f(x)$ have a unique global minimum at a point $x_u^*$ and $h(x_u^*) \gt 0$. I want to prove that i can find a global optimal point $x_c^*$ by solving convex problem of minimizing $f(x)$ subject to $h(x) \leq 0$.