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We investigate its properties, e.g.,\u00a0the Fenchel\u2013Young inequality and the characterization of the convex subdifferential using the analogue of the Fenchel\u2013Moreau Theorem. These properties of the Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions. Numerical results illustrate the performance of the algorithm, which competes with the recently derived Douglas\u2013Rachford algorithm on manifolds of nonpositive curvature. 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