Deciding uniqueness in norm maximization

NP-hardness is established for the problem whose instance is a system of linear inequalities defining a polytope P, and whose question is whether, on P, the global maximum of the Euclidean norm is attained at more than one vertex of P. The NP-hardness persists even for the restricted problem in which P is a full-dimensional parallelotope with one vertex at the origin. This makes it possible to establish NP-hardness for other uniqueness problems, including some from pseudoboolean programming and computational convexity.