Q:

why $\displaystyle\max_{k\in \{1,\cdots ,d\}} |a_k| \ge d^{-1}\sum_{k=1}^d |a_k|$?

Let $(a_k)_k$ be a sequence of complex numbers. Why
$$\displaystyle\max_{k\in \{1,\cdots ,d\}} |a_k|
\ge d^{-1} \displaystyle\sum_{k=1}^d |a_k|?$$
Thanks.

A:

for all $k=1,2\cdots, d$ $|a_k|\le \max |a_k|$
summing both you get
$$ \sum_{k=1}^d |a_k|\le \sum_{k=1}^d \max_{k\in \{1,\cdots ,d\}} |a_k| =d\max_{k\in \{1,\cdots ,d\}}|a_k|$$
that is $$\displaystyle\max_{k\in \{1,\cdots ,d\}} |a_k| \ge d^{-1}\sum_{k=1}^d |a_k|$$

