Monotonic norms and orthogonal issues in multidimensional voting

We study issue-by-issue voting by majority and incentive compatibility in multidimensional frameworks where privately informed agents have preferences induced by general norms and where dimensions are endogenously chosen. We uncover the deep connections between dominant strategy incentive compatibility (DIC) on the one hand, and several geometric/functional analytic concepts on the other. Our main results are: 1) Marginal medians are DIC if and only if they are calculated with respect to coordinates defined by a basis such that the norm is orthant-monotonic in the associated coordinate system. 2) Equivalently, marginal medians are DIC if and only if they are computed with respect to a basis such that, for any vector in the basis, any linear combination of the other vectors is Birkhoff-James orthogonal to it. 3) We show how semi-inner products and normality provide an analytic method that can be used to find all DIC marginal medians. 4) As an application, we derive all DIC marginal medians for l_p spaces of any finite dimension, and show that they do not depend on p (unless p=2).