For $\mathbf{x} \in \mathbb{R}^{n}$ we write $\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right)$. Define

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

(a) Suppose that $L$ is a convex subset of $P$, that $(1,1, \ldots, 1) \in L$ and that $\prod_{j=1}^{n} x_{j} \leqslant 1$ for all $\mathbf{x} \in L$. Show that $\sum_{j=1}^{n} x_{j} \leqslant n$ for all $\mathbf{x} \in L$.

(b) Suppose that $K$ is a non-empty closed bounded convex subset of $P$. Show that there is a $\mathbf{u} \in K$ such that $\prod_{j=1}^{n} x_{j} \leqslant \prod_{j=1}^{n} u_{j}$ for all $\mathbf{x} \in K$. If $u_{j} \neq 0$ for each $j$ with $1 \leqslant j \leqslant n$, show that

$$\sum_{j=1}^{n} \frac{x_{j}}{u_{j}} \leqslant n$$

for all $\mathbf{x} \in K$, and that $\mathbf{u}$ is unique.