Appendix Theorem 5 and nonunique minimizers

This is a proof-route correction, not a counterexample to the paper’s convergence claims.

The printed Appendix Theorem 5 (cited publication:1995-2035) assumes that the convex objective has a unique minimizer x* and concludes almost-sure convergence to that point. The proofs of Theorem 1 and Proposition 2 invoke that theorem. However, the main-text assumptions C1–C3 require only a bounded measurable density for ideal points. They do not imply uniqueness of the societal optimum or of a coordinatewise median.

For example, in one dimension let the feasible set be [-1, 1] and let the ideal point have density one on [-1, -1/2] ∪ [1/2, 1] and zero elsewhere. This is a bounded measurable density satisfying C3. The expected L1 cost has every point of [-1/2, 1/2] as a minimizer. Thus Appendix Theorem 5 cannot directly justify the (p,q) = (1,∞) and (∞,1) branches of Theorem 1, nor Proposition 2’s explicitly set-valued median conclusion, without an additional uniqueness argument.

The formalization proves the main conclusions using a minimizer-set argument:

1. For every minimizer, projected stochastic-subgradient descent makes the squared Euclidean distance to that minimizer converge almost surely.
2. Compactness and the objective-descent argument provide a subsequence whose limit is a minimizer.
3. The convergent distance potential to that subsequential minimizer forces the whole sample path to converge to it.

The almost-sure argument does not intersect events indexed by every minimizer. Finitely many fixed minimizers supply affine anchors: differences of their squared-distance potentials control the potential to any minimizer in their affine span. Together with a minimizer subsequence, this proves convergence of the whole path to a point in the minimizer set. The limit need not be a fixed minimizer chosen in advance.

This extension of Appendix Theorem 5 supplies the nonunique-minimizer cases in the main convergence results without adding a uniqueness assumption.