1. Given $x_{1}, x_{2}, \cdots, x_{2013} \in[-1,1]$. Then the minimum value of $x_{1} x_{2}+x_{2} x_{3}+\cdots+x_{2012} x_{2013}+x_{2013} x_{1}$ is ( ).
(A) -2013
(B) -2012
(C) -2011
(D) -2010
- 1. C.

If 2012 variables are fixed, the original expression becomes a linear function of one variable. Note that, when the variable takes 1 or -1, the function reaches its maximum value, hence $x_{1}, x_{2}, \cdots, x_{2013} \in\{-1,1\}$.

By trying, at least one of $x_{1} x_{2}, x_{2} x_{3}, \cdots, x_{2013} x_{1}$ is 1, thus the minimum value of the original expression is -2011, which can be achieved by
$$
\begin{array}{l}
x_{1}=x_{3}=x_{5}=\cdots=x_{2013}=1, \\
x_{2}=x_{4}=\cdots=x_{2012}=-1 .
\end{array}
$$