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We provide a simple proof for the Fenchel duality between strong convexityand Lipschitz continuous gradient. To this end, we first establish equivalentconditions of convexity for a general function that may not be differentiable.By utilizing these equivalent conditions, we can directly obtain equivalentconditions for strong convexity and Lipschitz continuous gradient. Based onthese results, we can easily prove Fenchel duality. Beside this main result, wealso identify several conditions that are implied by strong convexity orLipschitz continuous gradient, but are not necessarily equivalent to them. Thismeans that these conditions are more general than strong convexity or Lipschitzcontinuous gradient themselves.

On the Fenchel Duality between Strong Convexity and Lipschitz Continuous Gradient