Strategy-proof choice on strictly convex frontiers

A collective rule must often choose a public outcome on a constrained boundary. We characterize truthful choice with diagonal unanimity on the entire boundary of any compact full-dimensional strictly convex body in finite dimension at least two. On the full domain of linear support preferences, ordinary individual strategy-proofness and diagonal unanimity force the rule to select one fixed participant's preferred point at every profile. The population is any fixed finite nonempty set. No Pareto condition, boundary smoothness, or continuity or measurability of the rule is assumed. If interior outcomes are allowed instead, two or more participants can use positive fixed weighted averages of preferred points for truthful non-dictatorial compromise on the same type domain. The proof derives regularity from the two agents' incentive inequalities, transports a common support-measure bound across reports, and makes every convexified attainable menu a Minkowski summand of the feasible body. Boundary generation then forces fixed control. The result extends the spherical public-good location question and identifies the role of boundary-valued choice in eliminating truthful compromise.