Implementation of solutions to the problem of fair division when preferences are single-peaked

We consider the problem of fairly allocating an infinitely divisible commodity among a group of agents with single-peaked preferences. We address the issue of (Nash)-implementability of solutions to this problem. As the lack of monotonicity of preferences often causes solutions to violate no veto power, the classic Maskin-type theorems cannot be invoked. Instead, we show that more powerful theorems due to Danilov-Yamato and Sjostrom can help settle implementation questions for most of the central fairness concepts, generally in the positive.