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arxiv: 2604.04889 · v1 · submitted 2026-04-06 · 🧮 math.MG · math.CA· math.CO

Recognition: no theorem link

An improved bound for sumsets of thick compact sets via the Shapley--Folkman theorem

Scott Duke Kominers

Pith reviewed 2026-05-10 19:21 UTC · model grok-4.3

classification 🧮 math.MG math.CAmath.CO
keywords Minkowski sumsumset interiorFeng-Wu thicknessShapley-Folkman theoremcompact setsconvexificationMinkowski sum interior
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The pith

Compact sets with Feng-Wu thickness at least c in R^d have Minkowski sum with nonempty interior once n exceeds sqrt(d) over (sqrt(1+c) minus 1) squared.

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

The paper establishes an improved lower bound on the number of summands needed to guarantee that the Minkowski sum of compact sets with positive diameter and Feng-Wu thickness at least c has nonempty interior. Prior work required n to exceed a constant times c to the power of negative three; the new result shows that n larger than sqrt(d) times roughly one over c squared already suffices. The argument achieves this by replacing the earlier one-summand-at-a-time enlargement with a single simultaneous convexification step that invokes the radius form of the Shapley-Folkman theorem. For any fixed dimension the dependence on thickness therefore improves from cubic to quadratic.

Core claim

Let E1 through En be compact subsets of R^d each having positive diameter and Feng-Wu thickness at least c greater than zero. Then the sum E1 plus ... plus En has nonempty interior whenever n is larger than sqrt(d) divided by (sqrt(1 plus c) minus 1) squared, or equivalently larger than sqrt(d) times (sqrt(1 plus c) plus 1) squared over c squared. In particular the simpler bound n larger than 6 sqrt(d) over c squared works when 0 less than c less than or equal to 1. The proof obtains this by applying the radius form of the Shapley-Folkman theorem once to all summands simultaneously, which controls the distance from the sum to the sum of the convex hulls more efficiently than sequential one-s

What carries the argument

The radius form of the Shapley-Folkman theorem, which supplies an explicit bound on the distance between the sum of the sets and the sum of their convex hulls in terms of the number of nonconvex summands and their radii; this bound replaces the sequential enlargement used by Feng and Wu.

If this is right

  • For any fixed dimension the required number of summands grows only quadratically rather than cubically in the reciprocal of thickness.
  • The explicit bound n greater than 6 sqrt(d) over c squared suffices whenever thickness is at most one.
  • The exponent on the reciprocal of c improves from three to two at the sole cost of a multiplicative factor linear in the square root of dimension.

Where Pith is reading between the lines

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

  • The same simultaneous convexification step could be tested on other quantitative notions of thickness or on sums taken in normed spaces other than Euclidean space.
  • Low-dimensional numerical sampling of random thick compact sets might indicate whether the sqrt(d) prefactor can be removed or sharpened.
  • Related sumset problems that currently rely on sequential enlargement techniques may admit analogous quadratic improvements.

Load-bearing premise

The sets are compact with positive diameter and Feng-Wu thickness bounded below by some fixed positive c, so the radius form of the Shapley-Folkman theorem applies directly to control the convexification error for the entire sum at once.

What would settle it

Find explicit compact sets in some dimension d with Feng-Wu thickness exactly equal to a chosen c greater than zero such that their sum has empty interior for an integer n strictly larger than sqrt(d) over (sqrt(1 plus c) minus 1) squared.

read the original abstract

Let $E_1,\dots,E_n \subset \mathbb{R}^d$ be compact sets of positive diameter with Feng--Wu thickness at least $c>0$. Feng and Wu proved that $E_1+\cdots+E_n$ has non-empty interior when $n>2^{11}c^{-3}+1$. We show that \[n>\frac{\sqrt d}{(\sqrt{1+c}-1)^2}=\frac{\sqrt d\,(\sqrt{1+c}+1)^2}{c^2}\] already suffices. In particular, since $0<c\le 1$, the bound $n>6\sqrt d\,c^{-2}$ is enough. For fixed dimension $d$, this improves the exponent in $c^{-1}$ from $3$ to $2$, while introducing only an explicit factor of $\sqrt d$. The proof replaces the one-summand-at-a-time enlargement of Feng--Wu by a simultaneous convexification step based on a radius form of the Shapley--Folkman theorem.

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 / 2 minor

Summary. The manuscript proves that if E_1, …, E_n ⊂ R^d are compact sets of positive diameter each having Feng–Wu thickness at least c > 0, then the Minkowski sum E_1 + ⋯ + E_n has non-empty interior once n exceeds √d / (√(1+c) − 1)^2 (equivalently bounded by 6√d ⋅ c^{-2} for 0 < c ≤ 1). This improves the earlier Feng–Wu threshold n > 2^{11} c^{-3} + 1 by replacing the sequential one-summand enlargement with a single simultaneous convexification step that invokes a radius form of the Shapley–Folkman theorem; the thickness parameter c controls the relevant non-convexity radii after normalization to unit diameter.

Significance. The result supplies a quantitatively sharper, fully explicit threshold for when sums of uniformly thick compact sets acquire interior points. The reduction of the exponent on c from 3 to 2 (modulo an explicit √d factor) and the global application of Shapley–Folkman constitute a clean technical improvement over the prior sequential argument. The bound is parameter-free in the sense that it depends only on d and c, which makes it immediately usable in applications within additive combinatorics and geometric measure theory.

minor comments (2)
  1. [Abstract] Abstract: the simplified form n > 6√d ⋅ c^{-2} is stated for 0 < c ≤ 1; a one-line verification that (√(1+c) + 1)^2 / c^2 ≤ 6 on this interval would make the constant transparent.
  2. [Proof] The manuscript should confirm that the radius form of Shapley–Folkman is applied with the precise non-convexity radii induced by the Feng–Wu thickness condition after the unit-diameter normalization; a short display of the radius estimate would aid verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and encouraging report, which accurately summarizes the main result and its improvement over the prior Feng-Wu bound. We appreciate the recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central derivation applies an external radius form of the Shapley-Folkman theorem to the prior Feng-Wu thickness condition on compact sets of positive diameter. This yields the stated bound on n via simultaneous convexification, without any self-definitional steps, fitted parameters renamed as predictions, load-bearing self-citations, or ansatzes smuggled from the authors' prior work. The result is independent of the target interior-nonemptiness claim by construction and relies on externally verifiable theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the definition and properties of Feng-Wu thickness (taken from prior literature) and the standard Shapley-Folkman theorem; no free parameters are fitted and no new entities are introduced.

axioms (2)
  • domain assumption Feng-Wu thickness is well-defined for compact sets of positive diameter and satisfies the properties used in the enlargement argument
    Invoked to guarantee the sets are thick enough for the convexification step to produce interior.
  • standard math The radius form of the Shapley-Folkman theorem holds and can be applied simultaneously to the family of sets
    Cited as the replacement for the one-summand-at-a-time method of Feng-Wu.

pith-pipeline@v0.9.0 · 5486 in / 1487 out tokens · 57514 ms · 2026-05-10T19:21:11.518303+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. How Thick Is the Sierpi\'nski Triangle?

    math.MG 2026-05 unverdicted novelty 7.0

    The Feng-Wu thickness of the standard Sierpiński triangle of side length 1 is exactly √3/6.

Reference graph

Works this paper leans on

12 extracted references · cited by 1 Pith paper

  1. [1]

    Arrow and Frank H

    Kenneth J. Arrow and Frank H. Hahn, General Competitive Analysis , Holden-Day, San Francisco, CA, 1971

    1971

  2. [2]

    Reny, An improved bound for the S hapley-- F olkman theorem , Journal of Mathematical Economics 89 (2020), 48--52

    Eric Budish and Philip J. Reny, An improved bound for the S hapley-- F olkman theorem , Journal of Mathematical Economics 89 (2020), 48--52

    2020

  3. [3]

    Freiman, Foundations of a Structural Theory of Set Addition , Translations of Mathematical Monographs, vol

    Gregory A. Freiman, Foundations of a Structural Theory of Set Addition , Translations of Mathematical Monographs, vol. 37, American Mathematical Society, Providence, RI, 1973

    1973

  4. [4]

    1, 35--65

    De-Jun Feng and Yu-Feng Wu, On arithmetic sums of fractal sets in R ^d , Journal of the London Mathematical Society 104 (2021), no. 1, 35--65

    2021

  5. [5]

    4, 479--523

    Dmitry Kleinbock, Elon Lindenstrauss, and Barak Weiss, On fractal measures and D iophantine approximation , Selecta Mathematica (New Series) 10 (2004), no. 4, 479--523

    2004

  6. [6]

    Carlos Gustavo T. de A. Moreira and Jean-Christophe Yoccoz, Stable intersections of regular Cantor sets with large Hausdorff dimensions , Annals of Mathematics 154 (2001), no. 1, 45--96

    2001

  7. [7]

    58, American Mathematical Society, Providence, RI, 1987, pp

    Jacob Palis, Homoclinic orbits, hyperbolic dynamics and dimension of Cantor sets , The Lefschetz Centennial Conference, Part III ( Mexico City , 1984), Contemporary Mathematics, vol. 58, American Mathematical Society, Providence, RI, 1987, pp. 203--216

    1984

  8. [8]

    Ruzsa, Generalized arithmetical progressions and sumsets, Acta Mathematica Hungarica 65 (1994), no

    Imre Z. Ruzsa, Generalized arithmetical progressions and sumsets, Acta Mathematica Hungarica 65 (1994), no. 4, 379--388

    1994

  9. [9]

    a hle, eds.), Progress in Probability, vol. 70, Birkh \

    Pablo Shmerkin, Projections of self-similar and related fractals: a survey of recent developments, Fractal Geometry and Stochastics V (Christoph Bandt, Kenneth Falconer, and Martina Z \"a hle, eds.), Progress in Probability, vol. 70, Birkh \"a user/Springer, Cham, 2015, pp. 53--74

    2015

  10. [10]

    Starr, Quasi-equilibria in markets with non-convex preferences, Econometrica 37 (1969), no

    Ross M. Starr, Quasi-equilibria in markets with non-convex preferences, Econometrica 37 (1969), no. 1, 25--38

    1969

  11. [11]

    1, 93--104

    Hugo Steinhaus, Sur les distances des points dans les ensembles de mesure positive, Fundamenta Mathematicae 1 (1920), no. 1, 93--104

    1920

  12. [12]

    Haoyu Wu and Ao Tang, An even tighter bound for the S hapley-- F olkman-- S tarr theorem , Journal of Mathematical Economics 114 (2024), 103028

    2024