Integrals I: Feynman's Trick · Superchallenge problem

Real numbers $x_1, \ldots, x_n$ ($n \ge 3$) satisfy $x_i \le \dfrac{x_{i-1} + x_{i+1}}{2}$ for every $i$, indices taken mod $n$. What is the largest possible value of $\max x_i - \min x_i$?