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arxiv: 2606.30617 · v1 · pith:MBYEDMIJnew · submitted 2026-06-29 · 🧮 math.MG · math.PR

Peelings and Wrappings of Families of Convex Sets with Applications to Strongly Convex Sets Generated by Random Samples

Alexander Marynych , Mykyta Sadok This is my paper

Pith reviewed 2026-06-30 02:49 UTC · model grok-4.3

classification 🧮 math.MG math.PR
keywords convex setspeeling operationswrapping operationsrandom convex hullsPoisson point processpolaritystochastic geometry
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0 comments X

The pith

The m-point and recursive peelings of polar bodies from random K-hulls converge in distribution to those of a limiting Poisson object when K is strictly convex and regular.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines m-point peeling as the intersection of convex hulls obtained after deleting any m members of a family of compact convex sets, and recursive peeling as the result of repeatedly removing sets whose deletion changes the hull. It introduces the dual wrapping operations for intersections via polarity. A deterministic geometric framework establishes continuity of both procedures under vague convergence of locally finite point measures on the space of compact convex sets. In the probabilistic part, when K is strictly convex and regular, the peelings of the polar bodies of random K-hulls converge in distribution to the corresponding peelings of the limiting Poisson object; polarity transfers this to convergence of the wrapping operations on the rescaled random sets.

Core claim

Assuming K is strictly convex and regular, the m-point and recursive peelings of the polar bodies associated with the random K-hulls converge in distribution to the corresponding peelings of the limiting Poisson object; by polarity this yields distributional convergence of the associated wrapping operations for the rescaled random sets themselves.

What carries the argument

The m-point peeling (intersection of convex hulls after all possible deletions of m members) and recursive convex hull peeling (repeated removal of contributing sets), together with their polar duals (wrapping operations), shown continuous under vague convergence of point measures.

If this is right

  • Contributing sets are identifiable under general position assumptions on the family.
  • Compactness of convex hulls of subfamilies is required for the peeling constructions to be well-defined.
  • Both peeling procedures are continuous with respect to vague convergence of locally finite point measures.
  • The same distributional convergence holds for the dual wrapping operations on the rescaled random sets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The continuity results may permit passing to the limit in other functionals of the peeled families beyond the peeling itself.
  • The framework supplies a geometric language for studying depth and layers in finite point sets inside convex bodies.
  • Extensions to non-Poisson driving measures would require checking whether the vague-convergence continuity still applies.

Load-bearing premise

K must be strictly convex and regular.

What would settle it

Generate many large random samples from a convex body K that fails to be strictly convex, compute the empirical distribution of the m-point peelings of the polar bodies, and test whether it matches the distribution obtained from the Poisson limit; systematic deviation would falsify the claim.

read the original abstract

We introduce and study peeling and wrapping operations for families of compact convex sets. The two peeling procedures considered in the paper are the $m$-point peeling, obtained by intersecting the convex hulls remaining after all possible deletions of $m$ members of the family, and the recursive convex hull peeling, obtained by repeatedly removing the contributing sets, that is, those members whose deletion strictly changes the convex hull. Using polarity, we also introduce the dual wrapping operations for intersections of convex sets. The deterministic part of the paper develops the geometric framework needed for these constructions. In particular, we study contributing sets under general position assumptions, explain the role of compactness of convex hulls of subfamilies, and prove continuity results for both peeling procedures with respect to a suitable vague convergence of locally finite point measures on the space of compact convex sets. The probabilistic part applies this framework to $K$-hulls generated by random samples from a convex body $K$. Assuming that $K$ is strictly convex and regular, we prove that the m-point and recursive peelings of the polar bodies associated with the random $K$-hulls converge in distribution to the corresponding peelings of the limiting Poisson object. By polarity, this also yields distributional convergence of the associated wrapping operations for the rescaled random sets themselves.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces m-point peeling (intersection of convex hulls after deleting m members) and recursive convex hull peeling (repeated removal of contributing sets) for families of compact convex sets, along with dual wrapping operations obtained via polarity for intersections. The deterministic framework studies contributing sets under general position, compactness of sub-family hulls, and proves continuity of both peeling procedures with respect to vague convergence of locally finite point measures on the space of compact convex sets. The probabilistic application assumes K strictly convex and regular, and establishes that the m-point and recursive peelings of the polar bodies of random K-hulls converge in distribution to the corresponding peelings of the limiting Poisson object; polarity then yields distributional convergence for the wrapping operations on the rescaled random sets.

Significance. If the continuity and convergence results hold, the paper supplies a coherent geometric framework that transfers vague-convergence arguments to peeling and wrapping functionals on random convex sets. The polarity duality between peeling and wrapping is a clean technical device, and the application to K-hulls supplies explicit distributional limits that extend classical Poisson-process approximations in stochastic geometry. The work is technically self-contained once the general-position and compactness arguments are verified.

minor comments (3)
  1. [§2] The definition of vague convergence on the space of compact convex sets (used for the continuity theorems) should be stated explicitly in §2 rather than referenced only to prior literature, to make the deterministic part self-contained.
  2. [Theorem 5.3] In the statement of the main probabilistic theorem, the precise form of the rescaling for the random sets (before applying the wrapping convergence) is not fully spelled out in the abstract and should be written explicitly in the theorem statement.
  3. [§4] A short remark clarifying why strict convexity of K is needed for the general-position arguments in the Poisson limit (as opposed to mere convexity) would help readers distinguish the assumption from the regularity condition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the significance of the work, and the recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines peeling and wrapping operations on families of convex sets, develops a deterministic geometric framework using polarity and vague convergence of measures, and proves continuity results. The probabilistic convergence statements for random K-hulls invoke the explicit assumption that K is strictly convex and regular, then appeal to standard Poisson process limits from prior literature. No equation or claim reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation chain; the target distributional convergences are independent statements under stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the domain assumption that K is strictly convex and regular for the convergence to hold; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption K is strictly convex and regular
    Explicitly required for the distributional convergence of peelings and wrappings of random K-hulls.

pith-pipeline@v0.9.1-grok · 5767 in / 1212 out tokens · 41689 ms · 2026-06-30T02:49:42.150465+00:00 · methodology

discussion (0)

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Reference graph

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