Properties of the Fréchet Derivative of the Fenchel Conjugate of the Antidistance Function

The Fréchet derivative of the Fenchel conjugate of the antidistance function is locally Lipschitz in the Hilbert space setting. It is shown that the Gâteaux differential of the operator, at points of the existence, is bounded self-adjoint operator (it is symmetric everywhere defined on the Hilbert space). Assuming Gâteaux or Fréchet differentiability of the operator results on the strong convergence of subgradients along some arcs are also provided.