Set-valued conditional functionals of random sets
Pith reviewed 2026-05-22 19:04 UTC · model grok-4.3
The pith
Applying scalar gauges to the support function of a random closed convex set produces a set-valued extension of the gauge, including a conditional version.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Applying a gauge function to the support function of a random closed convex set X leads to a set-valued extension of the gauge. The conditional variant applies the gauge to the conditional distribution and produces a random closed convex set as its value; this specializes to the conditional set-valued quantile or conditional set-valued expectation of the random set.
What carries the argument
The set-valued gauge obtained by applying a scalar gauge to the support function of a random closed convex set.
Load-bearing premise
The random object is a closed convex set whose support function combines with the scalar gauge while preserving the gauge properties of normalization, positive homogeneity, monotonicity, and translation equivariance.
What would settle it
A concrete random closed convex set and gauge for which the constructed set-valued map fails to be monotone or translation equivariant.
read the original abstract
Many key quantities in statistics and probability theory such as the expectation, quantiles, expectiles and many risk measures are law-determined maps from a space of random variables to the reals. We call such a law-determined map, which is normalised, positively homogeneous, monotone and translation equivariant, a gauge function. Considered as a functional on the space of distributions, we can apply such a gauge to the conditional distribution of a random variable. This results in conditional gauges, such as conditional quantiles or conditional expectations. In this paper, we apply such scalar gauges to the support function of a random closed convex set $\bX$. This leads to a set-valued extension of a gauge function. We also introduce a conditional variant whose values are themselves random closed convex sets. In special cases, this functional becomes the conditional set-valued quantile or the conditional set-valued expectation of a random set. In particular, in the unconditional setup, if $\bX$ is a random translation of a deterministic cone and the gauge is either a quantile or an expectile, we recover the cone distribution functions studied by Andreas Hamel and his co-authors. In the conditional setup, the conditional quantile of a random singleton yields the conditional version of the half-space depth-trimmed regions.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript defines gauge functions as law-determined maps from distributions to reals that are normalized, positively homogeneous, monotone, and translation equivariant. It extends these to random closed convex sets by applying the scalar gauge to the support function, yielding a set-valued gauge whose values are claimed to be random closed convex sets. A conditional version is introduced whose outputs are random closed convex sets; special cases recover the conditional set-valued quantile, conditional set-valued expectation, and (unconditionally) the cone distribution functions of Hamel et al. when the random set is a translate of a deterministic cone.
Significance. If the central construction is valid, the paper supplies a unified definitional framework that extends scalar statistical functionals to the set-valued setting, recovering several known objects as special cases. This could facilitate further work on set-valued quantiles, risk measures, and depths for random sets. The recovery of existing concepts in special cases is a positive feature, though the work remains primarily definitional with limited new theorems or applications demonstrated.
major comments (2)
- [Definition of the set-valued gauge via support function] The construction (as described in the abstract and the support-function definition) sets h_{Φ(X)}(u) = gauge(h_X(u)). Positive homogeneity in u is preserved by the gauge axioms, but convexity in u is not. Convexity of h_X gives h_X(λu+(1-λ)v) ≤ λ h_X(u) + (1-λ) h_X(v); monotonicity of the gauge then yields gauge(h_X(λu+(1-λ)v)) ≤ gauge(λ h_X(u) + (1-λ) h_X(v)). For a nonlinear gauge (e.g., quantile or expectile) the right-hand side need not be ≤ λ gauge(h_X(u)) + (1-λ) gauge(h_X(v)), so the candidate support function can fail to be convex. This directly contradicts the claim that Φ(X) takes values in random closed convex sets and is load-bearing for the entire framework.
- [Conditional variant] The conditional construction inherits the same convexity issue: the conditional gauge applied pointwise to the conditional support function need not produce a convex function of u, so the output random set may not be closed and convex almost surely. The special-case recoveries (conditional quantile of a singleton, cone distribution functions) may hold because those gauges or set classes satisfy extra properties, but the general claim requires either a proof that convexity is preserved or an explicit restriction on the class of admissible gauges.
minor comments (2)
- [Abstract and introduction] The abstract and introduction could more clearly distinguish the unconditional set-valued gauge from the conditional version and state the precise measurability requirements on the random set X.
- [Notation and special cases] Notation for the random closed convex set (bold X) and its support function should be introduced once and used consistently; a short table of special cases (quantile, expectile, expectation) would improve readability.
Simulated Author's Rebuttal
We thank the referee for the thorough review and for highlighting a critical point regarding convexity preservation in our construction. We address each major comment below and will revise the manuscript to incorporate the necessary clarifications and restrictions.
read point-by-point responses
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Referee: [Definition of the set-valued gauge via support function] The construction (as described in the abstract and the support-function definition) sets h_{Φ(X)}(u) = gauge(h_X(u)). Positive homogeneity in u is preserved by the gauge axioms, but convexity in u is not. Convexity of h_X gives h_X(λu+(1-λ)v) ≤ λ h_X(u) + (1-λ) h_X(v); monotonicity of the gauge then yields gauge(h_X(λu+(1-λ)v)) ≤ gauge(λ h_X(u) + (1-λ) h_X(v)). For a nonlinear gauge (e.g., quantile or expectile) the right-hand side need not be ≤ λ gauge(h_X(u)) + (1-λ) gauge(h_X(v)), so the candidate support function can fail to be convex. This directly contradicts the claim that Φ(X) takes values in random closed convex sets and is load-bearing for the entire framework.
Authors: We acknowledge the validity of this concern. The referee correctly notes that the given axioms on the gauge do not automatically ensure convexity of u ↦ gauge(h_X(u)) for nonlinear gauges, due to the coupling of the random variables h_X(u) and h_X(v) through the underlying random set. In the revised manuscript we will introduce an additional axiom requiring the gauge to be convex in the sense that gauge(λY + (1-λ)Z) ≤ λ gauge(Y) + (1-λ) gauge(Z) for the relevant joint distributions arising from support functions. Under this strengthened definition we will prove that the resulting map is convex and sublinear, hence the support function of a closed convex set. We will also note that the special cases (expectation, cone distribution functions) satisfy the convexity requirement automatically. revision: yes
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Referee: [Conditional variant] The conditional construction inherits the same convexity issue: the conditional gauge applied pointwise to the conditional support function need not produce a convex function of u, so the output random set may not be closed and convex almost surely. The special-case recoveries (conditional quantile of a singleton, cone distribution functions) may hold because those gauges or set classes satisfy extra properties, but the general claim requires either a proof that convexity is preserved or an explicit restriction on the class of admissible gauges.
Authors: We agree that the conditional construction requires the same care. The revised manuscript will extend the convexity axiom to conditional gauges and provide an almost-sure argument that the conditional support function remains convex under the additional assumption. We will explicitly state that the recoveries of the conditional set-valued quantile and the cone distribution functions hold because the specific gauges or the deterministic-cone structure of the random sets ensure the convexity property is satisfied without further restrictions. This will be added as a dedicated remark following the main definition. revision: yes
Circularity Check
No significant circularity; construction is definitional and self-contained.
full rationale
The paper defines gauge functions as normalized, positively homogeneous, monotone, translation-equivariant maps and constructs the set-valued extension by pointwise application to the support function h_X(u), yielding h_{Φ(X)}(u) = gauge(h_X(u)). This is an explicit definition rather than a derivation that reduces to fitted inputs or self-referential equations. Special cases recover external results such as cone distribution functions (Hamel et al.) and conditional half-space depth-trimmed regions without load-bearing self-citation chains. The central claim remains independent of the paper's own fitted values or prior unverified assertions, qualifying as self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption A gauge function is normalised, positively homogeneous, monotone and translation equivariant.
- domain assumption The random object is a closed convex set.
invented entities (2)
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Set-valued gauge function
no independent evidence
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Conditional set-valued gauge
no independent evidence
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
We apply such scalar gauges to the support function of a random closed convex set X. This leads to a set-valued extension of a gauge function... G(X) = intersection H_w(g(h(X,w)))
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IndisputableMonolith/Foundation/RealityFromDistinction.leanreality_from_one_distinction unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
conditional set-valued gauge... largest A-measurable random closed convex set Y such that h(Y,W) ≤ g(h(X,W)|A)
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
discussion (0)
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