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 $p \in \Sigma_{N}$. (a) Suppose that $p_{1} = 0$ and $p_{2} > 0$. Show that there exists $p'$ in $\Sigma_{N}$ such that $J_{f}(p') > J_{f}(p)$ (you may look for $p'$ close to $p$). (b) Deduce that if $p \in \Sigma_{N}(f)$, then $p_{i} > 0$ for all $i \in \{1, \ldots, N\}$.
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 $p \in \Sigma_{N}$.
(a) Suppose that $p_{1} = 0$ and $p_{2} > 0$. Show that there exists $p'$ in $\Sigma_{N}$ such that $J_{f}(p') > J_{f}(p)$ (you may look for $p'$ close to $p$).
(b) Deduce that if $p \in \Sigma_{N}(f)$, then $p_{i} > 0$ for all $i \in \{1, \ldots, N\}$.