We study strategic candidate positioning in multidimensional spatial-voting elections. Voters and candidates are represented as points in Rᵈ, and each voter supports the candidate that is closest under a distance induced by an ₚ-norm. We prove that computing an optimal location for a new candidate is NP-hard already against a single opponent, whereas for a constant number of issues the problem is tractable: an O (n^d+1) hyperplane-enumeration algorithm and an O (n n) radial-sweep routine for d=2 solve the task exactly. We further derive the first approximation guarantees for the general multi-candidate case and show how our geometric approach extends seamlessly to positional-scoring rules such as k-approval and Borda. These results clarify the algorithmic landscape of multidimensional spatial elections and provide practically implementable tools for campaign strategy.
Cleveland et al. (Tue,) studied this question.