11. (20 points) For any 2016 complex numbers $z_{1}, z_{2}$, $\cdots, z_{2016}$, we have
$$
\sum_{k=1}^{2016}\left|z_{k}\right|^{2} \geqslant \lambda \min _{1 \leqslant k \leqslant 2016}\left\{\left|z_{k+1}-z_{k}\right|^{2}\right\},
$$

where, $z_{2017}=z_{1}$. Find the maximum value of $\lambda$.
11. Let $z_{k}=\left\{\begin{array}{ll}\frac{1}{2}, & 1 \leqslant k \leqslant 2016, k \text { is odd; } \\ -\frac{1}{2}, & 1 \leqslant k \leqslant 2016, k \text { is even. }\end{array}\right.$

Then $\sum_{k=1}^{2016}\left|z_{k}\right|^{2}=504$,
$$
\min _{1 \leqslant k \leqslant 2016}\left\{\left|z_{k+1}-z_{k}\right|^{2}\right\}=1 .
$$

Therefore, $\lambda \leqslant 504$.
We now prove that for any 2016 complex numbers $z_{1}, z_{2}$, $\cdots, z_{2016}$, we have
$$
\sum_{k=1}^{2016}\left|z_{k}\right|^{2} \geqslant 504 \min _{1 \leqslant k \leqslant 2016}\left\{\left|z_{k+1}-z_{k}\right|^{2}\right\} .
$$

If there exists a positive integer $k(1 \leqslant k \leqslant 2016)$ such that $\left|z_{k+1}-z_{k}\right|=0$, then
$$
\sum_{k=1}^{2016}\left|z_{k}\right|^{2} \geqslant 504 \min _{1 \leqslant k \leqslant 2016}\left\{\left|z_{k+1}-z_{k}\right|^{2}\right\}
$$

is clearly true.
Assume $\left|z_{k+1}-z_{k}\right| \neq 0$.
Then $\left|z_{k+1}-z_{k}\right|^{2}$
$$
\begin{array}{l}
=2\left(\left|z_{k}\right|^{2}+\left|z_{k+1}\right|^{2}\right)-\left|z_{k+1}+z_{k}\right|^{2} \\
\leqslant 2\left(\left|z_{k}\right|^{2}+\left|z_{k+1}\right|^{2}\right) . \\
\text { Hence } \sum_{k=1}^{2016}\left|z_{k}\right|^{2}=\frac{1}{2} \sum_{k=1}^{2016}\left(\left|z_{k}\right|^{2}+\left|z_{k+1}\right|^{2}\right) \\
\geqslant \frac{1}{4} \sum_{k=1}^{2016}\left|z_{k+1}-z_{k}\right|^{2} \\
\geqslant 504 \min _{1 \leqslant k \leqslant 2016}\left\{\left|z_{k+1}-z_{k}\right|^{2}\right\} .
\end{array}
$$

In conclusion, the maximum value of $\lambda$ is 504.