Let $x_{1}, x_{2}, \ldots, x_{n}$ be $d$-dimensional vectors of real numbers with $n$ sufficiently large but the exact value is not of importance. A function of $\mu$ is defined to be $$ \ell(\mu)=\sup \left\{\sum_{i=1}^{n} \log p_{i}: \sum_{i=1}^{n} p_{i} x_{i}=\mu ; \sum_{i=1}^{n} p_{i}=1, p_{1}>0, \ldots, p_{n}>0\right\} $$ on the space of the interior of the convex hull of $x_{1}, \ldots, x_{n}$.
(a) Show that this is a concave function of $\mu$ on the convex hull.
(b) Let $\bar{x}=n^{-1} \sum_{i=1}^{n} x_{i}$. Let $\mathbf{a}$ be a vector of length $d$. Prove that $\ell(\bar{x}+t \mathbf{a})$ is a decreasing function of $t$ when $t>0$. $$$$ Here I was trying to prove (b), but I was confused with the given condition, why do we need that $\mathbf{a}$ be a vector of length $d$? Why is this length associated with the dimension of vectors? And are there any references discussed the relevant problems before?
Because $\ell$ is concave, to show (b) it is enough to show that $\ell(\bar x)$ is a maximum. But taking $p_i=1/n$ we get $\sum_{i} p_i x_i=\bar x$ and $\sum_i \log p_i = -n\log n$. Now for any $\{ p_i\}$ such that $\sum_i p_i=1$ and $p_i>0$, we have \begin{align*} \sum_i \log p_i &= \sum_i \frac{n}{n} \log p_i\\ &\leq n\log\left( \sum_i \frac{1}{n} p_i \right)\\ &=-n\log n \end{align*} where the inequality is Jensen's inequality, using concavity of $\log$ and the fact that $\sum_i \frac{1}{n}=1$.
From there we can write for $t_1<t_2$, $\bar x+t_1 a = \left(1-\frac{t_1}{t_2}\right)\bar x + \frac{t_1}{t_2}(\bar x +t_2 a)$, applying concavity of $\ell$ yields (b).