Can strong duality holds for convex programs, where Slater's condition is not satisfied? I know that for a convex program, if Slater's condition is satisfied, then strong duality holds. I also know examples of convex programs, where Slater's condition is not satisfied and there is a positive duality gap. My question is: does there exist any convex program, which has zero duality gap but does not satisfy Slater's condition?

The linear programing problem
$$\begin{array}{l l} 
\text{min }       & \ \ \  0 \\ 
\text{subject to } & \ \ \ x & \leq 0 \\
                   & -x & \leq 0 
\end{array}$$
satisfies your requirements.
