Exact characterization for subdifferentials of a special optimal value function

For a closed set S and a bounded closed convex set U in a real normed vector space, we give exact subdifferential formulas of an optimal value function $$\mathrm {I}\!\Gamma _{S|U}$$ whose definition is based on the Minkowski function of U. $$\mathrm {I}\!\Gamma _{S|U}$$ covers distance function and indicator function as special cases. The main contribution is dropping two important assumptions of some main results in the literature.