Unique optimal partitions of distributions and connections to hazard rates and stochastic ordering

Scalar quantizer uniqueness (log-concave laws)

Use

DS15's assumption (S) — uniqueness of the optimal efficient-score interval quantizer — holds for Gaussian efficient scores by any of these; with Pollard (1981) this makes the population→empirical transfer of the scalar optimum fully rigorous in one dimension.

Does not transfer

one-dimensional and distortion-side only; uniqueness for $d_\psi>1$ fails in general (rotational symmetry), which is exactly why DS15 is stated for $d_\psi=1$.

Sources

Kieffer (1983, IEEE IT 29:42–47): log-concave density + convex, strictly increasing, $C^1$ error weight ⟹ the locally optimal $K$-level scalar quantizer is unique, hence globally optimal, and Lloyd iterations converge to it (secondary-verified via Kazıklı–Sarıtaş–Gezici–Linder–Yüksel, arXiv:2012.08265, Thm 3, and Liu & Pagès, JMLR 21(86), Lemma 10). Fleischer (1964, IEEE Conv. Rec. 12:104–111, verbatim): strict log-concavity ⟹ unique stationary point of the $N$-level MSE quantizer; Gaussian, Laplacian, Rayleigh covered explicitly. Trushkin (1982, IEEE IT 28:187–198): convex error weights (hypotheses unverified, cited by Liu–Pagès). Liu & Pagès Prop 11: strictly log-concave density ⟹ positive-definite distortion Hessian at the optimum — a published population-level conditioning statement at the scalar optimum. Mease & Nair (2006, Statistica Sinica 16:1299–1312, verbatim, 30 Aug 2026 audit): uniqueness under log-concavity of the density via likelihood-ratio ordering — and an explicit counterexample refuting Eubank (1988)'s weaker sufficient condition (log-concavity of $f(F^{-1}(u))$): a spliced exponential/Weibull density with three stationary solutions. Boundary for (S): the assumption must stay pinned to log-concavity of the density itself; Eubank-type weakenings are known-broken.