Let $f \in \mathbb{R}^{N}$ and $J_{f} : \Sigma_{N} \rightarrow \mathbb{R}$ defined by $J_{f}(p) = H_{N}(p) + \sum_{i=1}^{N} p_{i} f_{i}$. We denote $J_{f,*} = \sup\{J_{f}(p) \mid p \in \Sigma_{N}\}$ and $\Sigma_{N}(f) = \{p \in \Sigma_{N} \mid J_{f}(p) = J_{f,*}\}$.
Let $p \in \Sigma_{N}$. We now assume that $p_{i} > 0$ for all $i \in \{1, \ldots, N\}$. We denote $E_{0} = \{a \in \mathbb{R}^{N} \mid \sum_{i=1}^{N} a_{i} = 0\}$.
(a) Verify that $E_{0}$ is a vector subspace of $\mathbb{R}^{N}$ and give its dimension. Identify the orthogonal $E_{0}^{\perp}$ of $E_{0}$ for the canonical inner product on $\mathbb{R}^{N}$.
(b) Let $a \in E_{0}$ and $\tilde{p} : \mathbb{R} \rightarrow \mathbb{R}^{N}$ defined by $\tilde{p}(t) = p + ta$. Show that there exists $\epsilon > 0$ such that $\tilde{p}(t) \in \Sigma_{N}$ for all $t \in ]-\epsilon, \epsilon[$. Calculate the derivative of $\tilde{p}$ at 0.
(c) Suppose further that $p \in \Sigma_{N}(f)$. Show that for all $a \in E_{0}$, we have $\sum_{i=1}^{N} a_{i}(f_{i} - \ln(p_{i})) = 0$. Deduce that there exists $c \in \mathbb{R}$ such that $\ln(p_{i}) = f_{i} + c$ for all $i \in \{1, \ldots, N\}$.