  This paper is part of an emerging line of work at the intersection of machine
learning and mechanism design, which aims to avoid noise in training data by
correctly aligning the incentives of data sources. Specifically, we focus on
the ubiquitous problem of linear regression, where strategyproof mechanisms
have previously been identified in two dimensions. In our setting, agents have
single-peaked preferences and can manipulate only their response variables. Our
main contribution is the discovery of a family of group strategyproof linear
regression mechanisms in any number of dimensions, which we call generalized
resistant hyperplane mechanisms. The game-theoretic properties of these
mechanisms -- and, in fact, their very existence -- are established through a
connection to a discrete version of the Ham Sandwich Theorem.
