Write

$$P=\left\{\mathbf{x} \in \mathbb{R}^{n}: x_{j} \geqslant 0 \text { for all } 1 \leqslant j \leqslant n\right\}$$

and suppose that $K$ is a non-empty, closed, convex and bounded subset of $\mathbb{R}^{n}$ with $K \cap \operatorname{Int} P \neq \emptyset$. By taking logarithms, or otherwise, show that there is a unique $\mathbf{x}^{*} \in K \cap P$ such that

$$\prod_{j=1}^{n} x_{j} \leqslant \prod_{j=1}^{n} x_{j}^{*}$$

for all $\mathbf{x} \in K \cap P$.

Show that $\sum_{j=1}^{n} \frac{x_{j}}{x_{j}^{*}} \leqslant n$ for all $\mathbf{x} \in K \cap P$.

Identify the point $\mathbf{x}^{*}$ in the case that $K$ has the property

$$\left(x_{1}, x_{2}, \ldots, x_{n-1}, x_{n}\right) \in K \Rightarrow\left(x_{2}, x_{3}, \ldots, x_{n}, x_{1}\right) \in K$$

and justify your answer.

Show that, given any $\mathbf{a} \in \operatorname{Int} P$, we can find a set $K$, as above, with $\mathbf{x}^{*}=\mathbf{a}$.