Determination of convex functions via subgradients of minimal norm

We show, in Hilbert space setting, that any two convex proper lower semicontinuous functions bounded from below, for which the norm of their minimal subgradients coincide, they coincide up to a constant. Moreover, under classic boundary conditions, we provide the same results when the functions are continuous and defined over an open convex domain. These results show that for convex functions bounded from below, the slopes provide sufficient first-order information to determine the function up to a constant, giving a positive answer to the conjecture posed in Boulmezaoud et al. (SIAM J Optim 28(3):2049-2066, 2018) .ANID Chile under grant Fondecyt 1190110 1200283 3190229 ANID Chile under grant Fondecyt de Iniciacion 1118009