I need your help to go through the following mathematical steps. I am completely new to the rules used below and any help would be extremely appreciated.
Consider the function $G(\boldsymbol{x}): \mathbb{R}^J \rightarrow \mathbb{R}$, where $\boldsymbol{x}\equiv (x_1,..., x_J)$.
$G(\cdot )$ is known: give me any $\boldsymbol{x}$ and I can tell you the value $G(\boldsymbol{x})$.
We know that $G(\cdot )$ is convex.
Fix $\boldsymbol{x}=\boldsymbol{x}^*$ and let $$ (1) \hspace{1cm}\frac{\partial G(\boldsymbol{x}^*)}{\partial x_j}=a^*_j \text{ }\forall j=1,...,J $$ where $a^*_j\in \mathbb{R}$ is a (known) parameter $\forall j=1,...,J$ and $\boldsymbol{a}^*\equiv (a_1,..., a_J)$.
My objective is to recover $\boldsymbol{x}^*$ by "inverting" (1).
The source that I found uses the following steps:
A) Consider the function $\bar{G}(\boldsymbol{a}): \mathbb{R}^J\rightarrow \mathbb{R}$ prescribed by $$ \bar{G}(\boldsymbol{a})\equiv \begin{cases} \max_{\boldsymbol{\tilde{x}}} \Big(\sum_{j=1}^J \tilde{x}_j a_j- G(\boldsymbol{\tilde{x}}) \Big) & \text{if $\sum_{j=1}^Ja_j\leq 1$} \\ \infty & \text{otherwise} \end{cases} $$ This is the Legendre-Fenchel transform or convex conjugate of $G(\boldsymbol{x})$ and we know that $\bar{G}(\boldsymbol{a})$ is convex.
Since $G(\cdot )$ is known, also $\bar{G}(\cdot)$ is known.
B) Let $\boldsymbol{x}^{\text{opt}}(\boldsymbol{a}): \mathbb{R}^J\rightarrow \mathbb{R}^J$ be the function delivering the (unique?) solution of $$ \max_{\boldsymbol{\tilde{x}}} \Big(\sum_{j=1}^J \tilde{x}_j a_j- G(\boldsymbol{\tilde{x}}) \Big) $$ for every $\boldsymbol{a}\in \mathbb{R}^J$.
C) By the envelope theorem $$ \frac{\partial \bar{G}(\boldsymbol{a})}{\partial a_j}=x_j^{\text{opt}}(\boldsymbol{a}) \text{ }\forall j\in \{1,...,J\} $$
D) The source concludes that $$ \frac{\partial \bar{G}(\boldsymbol{a}^*)}{\partial a_j}=x_j^* \text{ }\forall j\in \{1,...,J\} $$
E) Since $\frac{\partial \bar{G}(\boldsymbol{a}^*)}{\partial a_j}$ is known $\forall j\in \{1,...,J\}$, we have recovered $\boldsymbol{x}^*$.
My doubts:
(i) How do we know that only $\boldsymbol{x}^*$ satisfies (1)?
(ii) Which properties of $G(\cdot)$ are sufficient to go through the steps above (convexity, strict convexity, C1,...)? And could you highlight where do we use those properties?
(iii) Where do we use the convexity (strict?) of $\bar{G}(\cdot)$? Do we need other properties of $\bar{G}(\cdot)$ to apply the envelope theorem?
(iv) How can we go from (C) to (D)?