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arxiv: 2606.10815 · v1 · pith:C6M75NLRnew · submitted 2026-06-09 · 🧮 math.MG · math.CO

The sharp threshold for Hausdorff convexification under Minkowski addition

Peter van Hintum This is my paper

Pith reviewed 2026-06-27 10:53 UTC · model grok-4.3

classification 🧮 math.MG math.CO
keywords Hausdorff distanceMinkowski additionconvex hullDyn-Farkhi conjectureiterated averagessharp thresholdcounterexamplecompact sets
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The pith

For every n at least 3 there exist compact sets in R^n whose distance to the convex hull stays fixed under the first n-1 Minkowski self-averages.

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

The paper shows that the symmetric version of the Dyn-Farkhi conjecture already fails in dimensions three and higher. It does so by exhibiting a compact set A for which the Hausdorff distance d to the convex hull satisfies d(A(k)) equals d(A) greater than zero for every iterated average up to k equals n minus one. This means repeated self-averaging leaves the non-convexity untouched until the nth step. The authors complement the construction with an explicit contraction bound that forces a drop after exactly n averages for any compact set.

Core claim

For every n greater than or equal to 3, there exists a compact set A subset R^n such that d(A(k)) equals d(A) greater than zero for every 1 less than or equal to k less than or equal to n minus 1, where d measures Hausdorff distance to the convex hull and A(k) is the k-fold Minkowski average. For every nonempty compact A in R^n with n at least 2, the same distance satisfies d(A(n)) less than or equal to one minus (n minus 1) over n times (2n minus 1) times d(A).

What carries the argument

The k-fold iterated Minkowski average A(k) equals one over k times the sum of k copies of A, used to track invariance of the Hausdorff distance d to the convex hull.

If this is right

  • The symmetric Dyn-Farkhi conjecture is false in every dimension n at least 3.
  • The threshold k equals n is sharp: the distance is guaranteed to decrease only after n averages.
  • The factor 1 minus (n-1) over n(2n-1) supplies an explicit worst-case contraction rate after n averages.
  • Repeated Minkowski averaging reduces non-convexity, but the reduction can be delayed by a dimension-dependent number of steps.

Where Pith is reading between the lines

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

  • The same sets may serve as test cases for other convexification operators beyond Minkowski averages.
  • The dimensional jump at n equals 3 suggests that symmetric and asymmetric averaging diverge only above a fixed threshold.
  • Numerical checks of the contraction constant in dimension 3 would give a direct test of sharpness.

Load-bearing premise

There exists a compact set in R^n for n at least 3 whose distance to its convex hull remains exactly constant under the first n-1 Minkowski self-averages.

What would settle it

An explicit compact set A in some dimension n at least 3 for which d(A(k)) is strictly smaller than d(A) for at least one k less than n, or a compact set in dimension n at least 2 whose d(A(n)) exceeds the stated fractional multiple of d(A).

Figures

Figures reproduced from arXiv: 2606.10815 by Peter van Hintum.

Figure 1
Figure 1. Figure 1: Construction of the set A in dimension n = 3 and the corresponding second Minkowski average A(2) = A+A 2 . Theorem 1.3. In any dimension n ≥ 3, there exists a compact set A ⊂ R n so that for all 1 ≤ k ≤ n − 1, we have d (A(k)) = d(A) > 0. This theorem almost complements a result by Fradelizi, Madiman, Marsiglietti, and Zvavitch [FMMZ18] (originally proved in [FMMZ16, Theorem 4] using a result from [Sch75])… view at source ↗
read the original abstract

The Dyn-Farkhi conjecture asserts that the square of the Hausdorff distance from a compact set to its convex hull is subadditive with respect to Minkowski addition. The conjecture is elementary in dimension 1, was recently proved by Meyer in dimension 2, and was disproved in dimensions $n\geq3$ by Fradelizi, Madiman, Marsiglietti, and Zvavitch. The symmetric case $A=B$, however, remained open. We show that the conjecture already fails in this restricted setting. More precisely, for every $n\geq3$, we construct a compact set $A\subset\mathbb{R}^n$ such that $$d(A(k))=d(A)>0$$ for every $1\leq k\leq n-1$, where $d(X)$ is the Hausdorff distance from $X$ to its convex hull and $A(k):=\frac1k (A+\dots+A)$ is the $k$-fold iterated Minkowski average of $A$. We also prove that the threshold $k=n$ is sharp: for every nonempty compact $A\subset\mathbb{R}^n$ with $n\geq 2$, we have $$d(A(n))\leq \left(1-\frac{n-1}{n(2n-1)}\right)d(A).$$

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 / 1 minor

Summary. The paper resolves the symmetric case of the Dyn-Farkhi conjecture by constructing, for every n≥3, a compact set A⊂R^n such that d(A(k))=d(A)>0 for 1≤k≤n-1, where d(X) denotes the Hausdorff distance from X to its convex hull and A(k) is the k-fold Minkowski average. It further proves that this threshold is sharp: for every nonempty compact A⊂R^n (n≥2), d(A(n))≤(1−(n−1)/(n(2n−1)))d(A).

Significance. If the claims hold, the work supplies an explicit geometric counterexample to the symmetric Dyn-Farkhi conjecture in dimensions n≥3 together with a parameter-free upper bound that applies to arbitrary compact sets and demonstrates sharpness of the threshold k=n. The constructive nature of the disproof and the direct, non-circular derivation of the inequality constitute clear strengths.

minor comments (1)
  1. The abstract states the bound for n≥2 but the construction is given only for n≥3; a brief remark in the introduction clarifying why the bound statement begins at n=2 would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and for recommending acceptance. The report accurately summarizes the main contributions.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation rests on an explicit geometric construction of a compact set A in R^n (n≥3) together with direct verification that d(A(k)) remains equal to d(A) for the first n-1 Minkowski averages, plus a separate parameter-free inequality bounding d(A(n)) that holds for arbitrary nonempty compact sets in Euclidean space. Neither part reduces to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain; the cited prior disproof in higher dimensions is by independent authors and is not invoked to justify the new construction or bound. The argument is therefore self-contained against external geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies exclusively on the standard axiomatic properties of the Hausdorff metric, Minkowski addition, and convex hulls in Euclidean space; no free parameters, new entities, or ad-hoc assumptions beyond those are visible in the abstract.

axioms (1)
  • standard math Standard properties of the Hausdorff metric and Minkowski addition on compact subsets of Euclidean space
    Foundational definitions and continuity properties assumed without proof in metric geometry.

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discussion (0)

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Reference graph

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