Is the center-outward median set a single point in dimension $d \ge 3$? (Removability of the origin singularity of the optimal-transport quantile map)

The center-outward quantile function $Q_\pm$ is the optimal transport map (for the quadratic cost) pushing the spherical uniform measure $U_d$ on the unit ball of $\mathbb{R}^d$, whose density $c_d|x|^{-(d-1)}$ blows up at the origin, onto a probability $P$ with density locally bounded away from zero and infinity. Figalli (2018) proved that $Q_\pm = \nabla\varphi$ is a homeomorphism from the punctured ball onto $\mathbb{R}^d \setminus K$, where $K$ (the subdifferential of $\varphi$ at the origin) is a compact convex set of Lebesgue measure zero, and that in dimension two $K = \{Q_\pm(0)\}$, so that $Q_\pm$ is a homeomorphism up to the origin and no genuine singularity appears. Whether $K$ must be a single point in dimension $d \ge 3$ — equivalently, whether $Q_\pm$ extends continuously to the origin and the center-outward median set reduces to one point — is open. Figalli states that he does not expect the dimension-two result to hold in $d \ge 3$, since the Monge–Ampère equation behaves worse in higher dimensions, but constructs no counterexample and declines to investigate, noting that a counterexample would not affect the existence of nested quantile contours. All subsequent regularity results (convex supports in 2020, non-convex supports in 2024, Hilbert-space analogues in 2026) hold on the punctured ball and explicitly leave the possible non-singleton nature of the median set unresolved; adjacent extensions of the classical regularity theory (unbounded domains, doubling measures, boundary blow-up) do not cover this interior-point singularity in $d \ge 3$.

Background

In dimension one, the distribution function $F$ of a probability $P$ ($F(z) = P((-\infty, z])$) and its inverse, the quantile function, underlie order statistics, ranks, and a large part of nonparametric inference; a key structural fact is that $2F - 1$ is the unique monotone map pushing $P$ forward to the uniform distribution on $[-1, 1]$, the unit ball of $\mathbb{R}$. For dimension $d \ge 2$ there is no canonical ordering of $\mathbb{R}^d$, and componentwise or depth-based multivariate generalizations of distribution and quantile functions lose the properties (equivariance, distribution-freeness of ranks) that make the univariate concepts inferentially useful.

To repair this, Chernozhukov, Galichon, Hallin and Henry (2017) and Hallin, del Barrio, Cuesta-Albertos and Matrán (2021) proposed a measure-transportation-based construction. The reference measure $U_d$, called the uniform measure over the unit ball (or spherical uniform), is the law of $R\cdot S$ with $R$ uniform on $[0,1]$ and $S$ uniform on the unit sphere, taken independent; by the change-of-variable formula it has density $u_d(x) = c_d |x|^{-(d-1)}$ restricted to the unit ball, where $c_d$ is the inverse of the surface area of the unit sphere. Note that this is not the Lebesgue-uniform measure on the ball: its density is unbounded at the origin, although it is integrable and satisfies $U_d(B_r(0)) = r$. The Brenier–McCann theorem of optimal transport states that, for the quadratic cost $|x-y|^2$, there is an essentially unique map pushing one absolutely continuous probability onto another which is the gradient of a convex function; the center-outward distribution function $F_\pm$ is defined as the (a.e. unique) gradient of a convex function pushing $P$ forward to $U_d$, and the center-outward quantile function $Q_\pm$ is its inverse, i.e., the gradient of the Legendre-dual convex potential $\varphi$ on the unit ball, mapping $U_d$ forward to $P$. Quantile regions and contours of order $\tau \in (0,1)$ are defined as the images under $Q_\pm$ of the closed ball of radius $\tau$ and of its boundary sphere, so that they have probability content $\tau$; for these to be closed, nested, and to have continuous boundaries, $Q_\pm$ must be a homeomorphism at least away from the origin, and the median set (the intersection of all quantile regions) equals the subdifferential of $\varphi$ at the origin, denoted $K$.

The analytic difficulty is that the potential satisfies, in the sense of Alexandrov (i.e., in the weak measure sense induced by the subdifferential of a convex function), a Monge–Ampère equation $\det D^2\varphi(x) = u_d(x)/p(\nabla\varphi(x))$ — a fully nonlinear elliptic equation whose right-hand side blows up at the interior point $0$. The classical regularity theory for such equations (what del Barrio, González-Sanz and Hallin (2020) call 'the classical regularity theory for solutions of Monge–Ampère equations associated with the name of Caffarelli') requires both densities to be bounded above and below on convex supports, so it does not apply in any neighborhood of the origin. Convex functions can also be non-differentiable: the subdifferential at a point (the set of slopes of its supporting hyperplanes) is a compact convex set, and it is a singleton exactly when the function is differentiable there. Thus the behavior of $Q_\pm$ at the origin is governed by the set $K$.

Figalli (2018) proved (Theorem 1.1) that if the density $p$ of $P$ satisfies $0 < \lambda_R \le p \le \Lambda_R$ inside every ball $B_R$, then $Q_\pm$ is a homeomorphism from the punctured ball $B_1 \setminus \{0\}$ onto $\mathbb{R}^d \setminus K$, where $K$ (the subdifferential of the potential at $0$) is a compact convex set of Lebesgue measure zero; that $Q_\pm$ is moreover a $C^{k+1,\alpha}$ diffeomorphism (respectively analytic) on the punctured ball when $p$ is locally $C^{k,\alpha}$ (respectively analytic); and, in part (c), that if $d = 2$ then $K = \{Q_\pm(0)\}$ and $Q_\pm$ is a homeomorphism from the unit ball onto $\mathbb{R}^2$, so that in the plane the origin is not a genuine singularity. The proof of part (c) is intrinsically two-dimensional: it assumes by contradiction that $K$ contains a nondegenerate segment and derives a contradiction from the planar geometry of sections of convex functions. In Remark 1.2, Figalli notes that the Monge–Ampère equation behaves better in dimension two than in higher dimensions, states that he does not expect Theorem 1.1(c) to be true in dimension $d \ge 3$, but also that finding a counterexample would not be relevant to the existence of quantile contours (which follows from the homeomorphism on the punctured ball), so the question is left uninvestigated.

Subsequent work has extended the punctured-ball regularity but not reached the origin. del Barrio, González-Sanz and Hallin (2020) proved the same homeomorphism on the punctured ball for probabilities with bounded or unbounded convex support and observed that $Q_\pm$ may then fail to be continuous (single-valued) at the origin, all points of $K$ being mapped by $F_\pm$ to $0$. Hallin, del Barrio, Cuesta-Albertos and Matrán (2021) built the inferential theory on this basis and formalized the median set, noting that continuity of $Q_\pm$ at the origin is impossible unless the subdifferential at $0$ is a singleton. del Barrio and González-Sanz (2024) proved continuity on the interior of possibly non-convex supports and wrote that the median set is 'a compact convex negligible set, but possibly not a singleton, which means that the continuity of $Q_\pm$ requires additional assumptions'. Adjacent extensions of the regularity theory — monotone transport maps between unbounded domains (Cordero-Erausquin and Figalli 2019), Brenier potentials between locally doubling measures (Figalli and Jhaveri 2023), quantitative stability for reference densities blowing up near a domain boundary (Letrouit and Mérigot 2024) — do not cover an unbounded density at an interior point, and the discretized/empirical setting studied by Ghosal and Sen (2022) and the Hilbert-space analogue of González-Sanz, Hallin and Sen (2026) do not settle the population-level question either.

Problem Statement

Let $d \ge 3$, and let $U_d$ be the spherical uniform probability measure on the closed unit ball of $\mathbb{R}^d$, with density $u_d(x) = c_d |x|^{-(d-1)}$ restricted to the unit ball, where $c_d$ is the inverse of the surface area of the unit sphere ($U_d$ is the law of $R\cdot S$ with $R$ uniform on $[0,1]$ and $S$ uniform on the sphere). Let $P = p(y)\,dy$ be a probability measure on $\mathbb{R}^d$ whose density satisfies $0 < \lambda_R \le p \le \Lambda_R$ inside every ball $B_R$ of finite radius (Figalli's class; the equivalent variant in which $P$ is supported on a bounded or unbounded convex set with $p$ locally bounded away from zero and infinity on that support, as in del Barrio, González-Sanz and Hallin (2020), is equally open; it suffices to answer for either of the two variants, declaring which one is used). Let $\varphi$ be the Brenier potential, unique up to an additive constant, whose gradient $Q_\pm = \nabla\varphi$ is the optimal transport map for the quadratic cost pushing $U_d$ forward to $P$, and let $K$ be the subdifferential of $\varphi$ at the origin — a compact convex set of Lebesgue measure zero, which is the center-outward median set (the intersection of all center-outward quantile regions of positive order).

Determine whether $K$ is necessarily a single point. That is, either: (i) prove that for every $d \ge 3$ and every $P$ in the stated class, $\varphi$ is differentiable at the origin, so that $K = \{\nabla\varphi(0)\}$, the quantile map $Q_\pm$ extends continuously (indeed, as in dimension two, to a homeomorphism from the unit ball onto its image), and the center-outward median set consists of exactly one point; or (ii) exhibit, for some $d \ge 3$, a density $p$ in the stated class for which $K$ is a nontrivial compact convex set (for instance, one containing a nondegenerate line segment), so that the center-outward quantile map has a genuine singularity at the origin and the center-outward median set is not a single point.

Figalli (2018, Remark 1.2) states that he does not expect the dimension-two conclusion of Theorem 1.1(c) to hold in $d \ge 3$ — that is, he conjectures outcome (ii) — but leaves the question open. A resolution in either direction is explicitly noted in the source to leave the existence of closed, nested quantile contours (which require only the punctured-ball homeomorphism) unaffected; the question concerns the structure of the quantile map and the median at the center in general dimension.

References

1. Alessio Figalli, On the continuity of center-outward distribution and quantile functions, Nonlinear Analysis 177 (2018), part B, 413-421. DOI: 10.1016/j.na.2018.05.008, https://doi.org/10.1016/j.na.2018.05.008 (preprint arXiv:1805.04946, https://arxiv.org/abs/1805.04946)
2. Eustasio del Barrio, Alberto González-Sanz, Marc Hallin, A note on the regularity of optimal-transport-based center-outward distribution and quantile functions, Journal of Multivariate Analysis 180 (2020), 104671. DOI: 10.1016/j.jmva.2020.104671, https://doi.org/10.1016/j.jmva.2020.104671 (preprint arXiv:1912.10719, https://arxiv.org/abs/1912.10719)
3. Marc Hallin, Eustasio del Barrio, Juan Cuesta-Albertos, Carlos Matrán, Distribution and quantile functions, ranks and signs in dimension d: a measure transportation approach, The Annals of Statistics 49 (2021), no. 2, 1139-1165. DOI: 10.1214/20-AOS1996, https://doi.org/10.1214/20-AOS1996 (preprint arXiv:1806.01238, https://arxiv.org/abs/1806.01238)
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5. Marc Hallin, Dimitri Konen, Multivariate Quantiles: Geometric and Measure-Transportation-Based Contours, in: Applications of Optimal Transport to Economics and Related Topics, Studies in Systems, Decision and Control, Springer, 2024, pp. 61-78. DOI: 10.1007/978-3-031-67770-0_6, https://doi.org/10.1007/978-3-031-67770-0_6 (preprint arXiv:2401.02499, https://arxiv.org/abs/2401.02499)
6. Alessio Figalli, Yash Jhaveri, Regularity properties of monotone measure-preserving maps, Advanced Nonlinear Studies 23 (2023), no. 1, article 20220057. DOI: 10.1515/ans-2022-0057, https://doi.org/10.1515/ans-2022-0057 (preprint arXiv:2305.09001, https://arxiv.org/abs/2305.09001)
7. Victor Chernozhukov, Alfred Galichon, Marc Hallin, Marc Henry, Monge-Kantorovich depth, quantiles, ranks and signs, The Annals of Statistics 45 (2017), no. 1, 223-256. DOI: 10.1214/16-aos1450, https://doi.org/10.1214/16-aos1450
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