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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?

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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.

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