Consider a convex optimisation problem in the standard form: $\min f_0(\mathbf x)$ subject to $f_i(\mathbf x)\le 0$ and $A\mathbf x=\mathbf b$, with $f_i$ all convex functions.
When discussing Slater's condition, the Wikipedia page mentions (if I understand it correctly) that Slater's conditions state that strong duality holds if there's a point $\mathbf x^*$ in the relative interior of the intersection of the domains of the constraint functions corresponding to which all inequality constraints are satisfied with strict inequality.
Why do we need a point in the relative interior of the domain here? What fails if we just ask for an interior point? Some simple examples would probably help to make this point clear.