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) .

中文翻译：

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通过最小范数的次梯度确定凸函数

我们证明，在希尔伯特空间设置中，任何两个从下方有界的凸真下半连续函数，对于它们的最小次梯度的范数重合，它们重合到一个常数。此外，在经典边界条件下，当函数是连续的并定义在开放凸域上时，我们提供相同的结果。这些结果表明，对于从下方有界的凸函数，斜率提供了足够的一阶信息来确定函数直到一个常数，对 Boulmezaoud 等人提出的猜想给出了肯定的答案。(SIAM J Optim 28(3):2049–2066, 2018)。