arxiv: 2603.17726 · v3 · submitted 2026-03-18 · 🧮 math.AP · math.MG· math.OC

Recognition: 2 theorem links

· Lean Theorem

Quantitative Stability for Minkowski's problem

K\'aroly B\"or\"oczky , Jo\~ao Miguel Machado , Jo\~ao P. G. Ramos

Authors on Pith no claims yet

Pith reviewed 2026-05-15 08:51 UTC · model grok-4.3

classification 🧮 math.AP math.MGmath.OC

keywords Minkowski problemquantitative stabilityconvex bodiesHausdorff distanceFraenkel asymmetrysurface area measuresdual-convex distanceL_p-Minkowski bodies

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The pith

Minkowski bodies from nearby sphere measures differ by at most the dual-convex distance to the power 1 over n minus 1 in Hausdorff distance.

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

The paper establishes quantitative stability for Minkowski bodies by controlling how close two such bodies must be when their generating measures on the sphere are close. For probability measures satisfying quantitative dispersion assumptions, the Hausdorff distance between the bodies, after optimal translation, is bounded by a constant times the dual-convex distance raised to 1 over n minus 1. The square of the Fraenkel asymmetry between the bodies obeys a stronger bound with exponent 1 plus 1 over n minus 1. These rates follow from analyzing a variational problem whose optimizers are the Minkowski bodies together with quantitative Brunn-Minkowski and isoperimetric inequalities. The results also cover L_p-Minkowski bodies for p not equal to n, and the Hausdorff exponent is shown to be sharp.

Core claim

For every pair of probability measures μ, ν satisfying a quantitative form of the classical dispersion assumptions yielding existence of Minkowski bodies, inf over x of the Hausdorff distance between E_μ and x plus E_ν is at most C times d_C(μ,ν) to the power 1 over n minus 1, while the square of the Fraenkel asymmetry α(E_μ, E_ν) is at most C times d_C(μ,ν) to the power 1 plus 1 over n minus 1. The same form of control holds for the L_p-Minkowski bodies when 1 ≤ p ≠ n. The exponent in the Hausdorff bound is sharp.

What carries the argument

The dual-convex distance d_C between probability measures on the sphere, which enters a variational problem whose optimizers are the Minkowski bodies E_μ and whose strong-concavity properties are obtained from quantitative Brunn-Minkowski and isoperimetric inequalities.

If this is right

The solution map from measures to Minkowski bodies is Hölder continuous with explicit exponent 1 over n minus 1 in the Hausdorff metric.
The same stability holds for L_p-Minkowski bodies in the range 1 ≤ p ≠ n.
The Fraenkel asymmetry between bodies decays at the faster rate given by the squared bound.
The sharpness example for the Hausdorff exponent shows that no better power is possible in general dimension.

Where Pith is reading between the lines

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

The variational approach and quantitative inequalities used here could supply error rates for numerical reconstruction of convex bodies from discrete or noisy surface-area data.
The same stability technique may extend to related inverse problems such as the Christoffel problem or other Minkowski-type problems with different curvature measures.
In two dimensions the stronger asymmetry bound might improve convergence rates for iterative schemes that recover planar convex sets from their support functions or widths.

Load-bearing premise

The measures satisfy a quantitative version of the classical dispersion assumptions that guarantee existence of the Minkowski bodies.

What would settle it

A sequence of measure pairs whose dual-convex distance tends to zero while the Hausdorff distance between the corresponding Minkowski bodies stays larger than any multiple of that distance to the power 1 over n minus 1 would falsify the claimed bound.

read the original abstract

We derive quantitative stability results for Minkowski bodies, as well as their counterparts, the $L_p$-Minkowski bodies in the range $1 \le p \neq n$. We prove that, for every pair of probability measures $\mu,\nu$ satisfying a quantitative form of the classical dispersion assumptions yielding existence of such bodies, we have a control of the form \[ \inf_{x\in \mathbb{R}^n}\mathrm{d_H}(E_\mu, x + E_\nu) \le C \mathrm{d_C}(\mu,\nu)^{\frac{1}{n-1}}, \quad \alpha(E_\mu, E_\nu)^2 \le C \mathrm{d_C}(\mu,\nu)^{1 + \frac{1}{n-1}}, \] where $\mathrm{d_H}$ denotes the Hausdorff distance, $\alpha$ denotes the Fraenkel asymmetry and $\mathrm{d_C}$ is the dual-convex distance of probability measures on the sphere. Our arguments are based on a variational problem whose optimizers are Minkowski bodies, for which we can obtain strong-concavity properties with the quantitative Brunn-Minkowski and isoperimetric inequalities. While the exponent in the Hausdorff distance is sharp, the exponent in the Fraenkel asymmetry is optimal in dimension $2$.

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.

Desk Editor's Note private letter to a colleague

This paper gives explicit quantitative stability for Minkowski bodies with the exponent 1/(n-1) on Hausdorff distance via variational strong concavity.

read the letter

The paper's main contribution is a pair of quantitative stability inequalities for Minkowski bodies and L_p-Minkowski bodies. For measures mu and nu close in the dual-convex distance d_C, the Hausdorff distance between the corresponding bodies is at most C times d_C to the power 1/(n-1), and the squared Fraenkel asymmetry is controlled by C times d_C to the power 1 plus 1/(n-1). The first exponent is sharp, and the second is optimal in dimension 2. They achieve this by setting up a variational problem whose solutions are the Minkowski bodies, then using quantitative versions of Brunn-Minkowski and isoperimetric inequalities to establish strong concavity of the functional. From there the stability follows directly from the modulus of concavity and the relation to the dual-convex distance. The argument covers both the classical case and L_p for p not equal to n, and they check optimality in low dimensions for the asymmetry part. This is a clean application of existing quantitative inequalities to a classical existence problem. The assumptions are the standard dispersion conditions made quantitative, so no new restrictions. The constants C come from those inequalities rather than being fitted. One potential soft spot is that the strong-concavity step needs careful checking for the precise dependence, especially in higher dimensions, but the outline shows no inconsistency in the exponents. The paper stays within convex geometry and does not claim broader impact. This work is aimed at researchers in convex geometry who need explicit rates for stability or approximation of convex bodies. A reader working on related problems in geometric inequalities or Minkowski-type problems would get direct value from the exponents and the method. It is a focused, technically competent piece that advances the quantitative side of a classical topic. I would send it to peer review. The claims are precise, the novelty is clear from the exponents, and the argument rests on solid ground.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes quantitative stability for Minkowski bodies E_μ and their L_p counterparts (1 ≤ p ≠ n) associated to probability measures μ, ν on the sphere. Under a quantitative version of the classical dispersion assumptions guaranteeing existence, it proves inf_x d_H(E_μ, x + E_ν) ≤ C d_C(μ,ν)^{1/(n-1)} and α(E_μ, E_ν)^2 ≤ C d_C(μ,ν)^{1 + 1/(n-1)}, where d_H is Hausdorff distance, α is Fraenkel asymmetry, and d_C is the dual-convex distance. The proof proceeds via a variational problem whose optimizers are the Minkowski bodies, combined with strong concavity obtained from quantitative Brunn-Minkowski and isoperimetric inequalities. The Hausdorff exponent is stated to be sharp, while the asymmetry exponent is optimal in dimension 2.

Significance. If the derivations hold, the results supply sharp quantitative stability estimates for the Minkowski problem in the measure-theoretic setting, extending classical existence theorems with explicit rates in terms of d_C. The variational characterization and direct use of quantitative Brunn-Minkowski/isoperimetric inequalities to obtain the concavity modulus constitute a clean approach that yields the claimed exponents without hidden losses. This contributes a useful stability theory to convex geometry and related PDEs, with the sharpness statement for the Hausdorff bound being a notable strength.

minor comments (3)

In the abstract and §1, the dual-convex distance d_C(μ,ν) is used without an explicit definition or reference to its precise formula; adding a short inline description or pointer to the definition in §2 would improve readability for readers outside the immediate subfield.
The dependence of the constant C on dimension n and on the quantitative dispersion parameters is not tracked explicitly in the statement of the main theorem; a remark clarifying whether C remains uniform under the stated assumptions would strengthen the result.
Figure 1 (if present) or the illustrative examples in §4 comparing the two exponents would benefit from a caption that explicitly recalls the relation between d_C and the classical dispersion quantities.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition of the clean variational approach and the sharpness of the Hausdorff exponent. The recommendation for minor revision is noted, and we will prepare a revised version incorporating any editorial suggestions.

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent standard inequalities

full rationale

The paper's central stability estimates follow from a variational formulation whose strong-concavity modulus is obtained directly from the quantitative Brunn-Minkowski and isoperimetric inequalities. These inequalities are classical results whose proofs are external to the present work and do not depend on the Minkowski bodies E_μ, E_ν or the dual-convex distance d_C. The exponents 1/(n-1) and 1 + 1/(n-1) arise mechanically from the concavity deficit and the relation between functional deficit and d_C; no parameter is fitted to the target data, no self-citation supplies a load-bearing uniqueness theorem, and no step reduces by definition to its own input. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The result rests on the existence theory for Minkowski bodies under dispersion assumptions and on quantitative versions of the Brunn-Minkowski and isoperimetric inequalities; no new free parameters or invented entities are introduced beyond the universal constant C whose dependence is left implicit.

free parameters (1)

C
Universal constant whose value depends on dimension and the dispersion constants of the measures; not fitted to the stability data itself.

axioms (2)

domain assumption Quantitative dispersion assumptions on the probability measures guarantee existence of the Minkowski bodies
Invoked to ensure E_μ and E_ν exist before stability can be discussed.
standard math Quantitative Brunn-Minkowski and isoperimetric inequalities hold with explicit constants
Used to obtain strong concavity of the variational functional.

pith-pipeline@v0.9.0 · 5552 in / 1443 out tokens · 39687 ms · 2026-05-15T08:51:22.414023+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Our arguments are based on a variational problem whose optimizers are Minkowski bodies, for which we can obtain strong-concavity properties with the quantitative Brunn-Minkowski and isoperimetric inequalities.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1 … α(E_μ, E_ν)² ≤ C d_C(μ,ν)^{1 + 1/n}

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