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arxiv: 2605.28262 · v2 · pith:T2PJUAZLnew · submitted 2026-05-27 · 🧮 math.AP

L_p Minkowski problem and Brunn-Minkowski inequality for dual quermassintegrals

Xiaojuan Chen , Shengyu Tang , Sinan Wang This is my paper

Pith reviewed 2026-06-29 11:17 UTC · model grok-4.3

classification 🧮 math.AP
keywords L_p dual Minkowski problemBrunn-Minkowski inequalitydual quermassintegralsconvex bodiesC^0 estimatesuniquenessorigin-symmetric
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The pith

C^0 estimates hold for the L_p dual Minkowski problem when 0 < p < q ≤ n without symmetry assumptions on the bodies.

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

The paper seeks to resolve the L_p dual Minkowski problem by obtaining C^0 estimates on solutions in the range 0 < p < q ≤ n, dropping any requirement that the convex bodies be symmetric. It establishes uniqueness of smooth solutions whenever the given density function lies sufficiently close to a constant in the Hölder norm. These facts are then applied, via the known link between uniqueness and the Brunn-Minkowski inequality, to prove the L_p Brunn-Minkowski inequality for dual quermassintegrals of origin-symmetric convex bodies when p < q. A reader would care because these estimates and inequalities control how volumes and surface measures interact for convex sets under scaling and addition operations.

Core claim

For 0 < p < q ≤ n the L_p dual Minkowski problem admits C^0 estimates on its solutions without symmetry assumptions on the bodies. Smooth solutions are unique provided the density function is sufficiently close to a constant in the Hölder norm. Exploiting the equivalence of uniqueness to the Brunn-Minkowski inequality, the L_p Brunn-Minkowski inequality holds for dual quermassintegrals of origin-symmetric convex bodies when p < q.

What carries the argument

The L_p dual Minkowski problem, which requires a convex body whose L_p dual curvature measure equals a prescribed measure, together with the equivalence between uniqueness of its solutions and validity of the associated Brunn-Minkowski inequality.

If this is right

  • C^0 estimates are obtained for the L_p dual Minkowski problem in the range 0 < p < q ≤ n without symmetric assumptions.
  • Smooth solutions are unique when the density is sufficiently close to a constant in the Hölder norm.
  • The L_p Brunn-Minkowski inequality for dual quermassintegrals holds for origin-symmetric convex bodies when p < q.

Where Pith is reading between the lines

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

  • The C^0 estimates could serve as the first step toward existence proofs via approximation or continuity methods.
  • The uniqueness-equivalence technique might extend to other dual curvature problems in convex geometry.
  • The restriction to origin-symmetric bodies for the inequality indicates that non-symmetric cases may need separate arguments.

Load-bearing premise

Uniqueness of solutions to the Minkowski-type problem is equivalent to the Brunn-Minkowski inequality holding in a certain sense.

What would settle it

A density function close to a constant in the Hölder norm for which the L_p dual Minkowski problem admits two distinct smooth solutions would falsify the uniqueness statement.

read the original abstract

This paper studies the core problems in the $L_p$ dual Brunn-Minkowski theory, encompassing the $L_p$ Minkowski problem and $L_p$ Brunn-Minkowski inequality for dual quermassintegrals. For the case $0<p<q\leq n$, we establish $C^0$ estimates for the $L_p$ dual Minkowski problem without symmetric assumptions, thereby resolving a related problem proposed by B\"or\"oczky-Chen-Liu-Saroglou in the smooth sense. We further prove the uniqueness of smooth solutions under appropriate conditions, provided the density function is sufficiently close to a constant in the H\"older norm. Finally, exploiting the fact that the uniqueness of the Minkowski type problem is equivalent to the validity of the Brunn-Minkowski inequality in a certain sense, we study the $L_p$ Brunn-Minkowski inequality for dual quermassintegrals for origin-symmetric convex bodies with $p<q$.

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

2 major / 1 minor

Summary. The paper claims to establish C^0 estimates for the L_p dual Minkowski problem for 0 < p < q ≤ n without symmetric assumptions on the data, thereby resolving a related problem proposed by Böröczky-Chen-Liu-Saroglou in the smooth sense. It further proves uniqueness of smooth solutions when the density function is sufficiently close to a constant in the Hölder norm. Finally, by exploiting the equivalence between uniqueness of Minkowski-type problems and the validity of the Brunn-Minkowski inequality in a certain sense, the paper studies the L_p Brunn-Minkowski inequality for dual quermassintegrals for origin-symmetric convex bodies with p < q.

Significance. If the C^0 estimates hold without symmetry assumptions, the work would constitute a meaningful advance in the L_p dual Brunn-Minkowski theory by addressing an open question in the smooth category. The uniqueness result under a Hölder-closeness condition and the subsequent derivation of the inequality would add value, provided the invoked equivalence applies directly to dual quermassintegrals in the stated range.

major comments (2)
  1. [Abstract] Abstract (final paragraph): the derivation of the L_p Brunn-Minkowski inequality for dual quermassintegrals rests on the statement that 'the uniqueness of the Minkowski type problem is equivalent to the validity of the Brunn-Minkowski inequality in a certain sense.' No reference, prior citation, or self-contained verification is supplied for this equivalence when applied to dual quermassintegrals with 0 < p < q ≤ n. Because this equivalence is load-bearing for the inequality claim, its applicability in the present setting must be justified explicitly.
  2. [Uniqueness part] Uniqueness statement: uniqueness of smooth solutions is obtained only under the additional hypothesis that the density is sufficiently close to a constant in the Hölder norm. The paper should clarify whether this restriction is necessary for the C^0 estimates themselves or only for the uniqueness step, and how it affects the resolution of the Böröczky-Chen-Liu-Saroglou question.
minor comments (1)
  1. [Notation] Notation for dual quermassintegrals and the associated L_p functionals should be introduced once and used consistently; any redefinition in later sections risks confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the paper accordingly to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final paragraph): the derivation of the L_p Brunn-Minkowski inequality for dual quermassintegrals rests on the statement that 'the uniqueness of the Minkowski type problem is equivalent to the validity of the Brunn-Minkowski inequality in a certain sense.' No reference, prior citation, or self-contained verification is supplied for this equivalence when applied to dual quermassintegrals with 0 < p < q ≤ n. Because this equivalence is load-bearing for the inequality claim, its applicability in the present setting must be justified explicitly.

    Authors: We agree that the equivalence requires explicit justification in this context. Although the equivalence is standard in the Minkowski problem literature, we will add a brief self-contained argument or appropriate reference tailored to the dual quermassintegrals case for 0 < p < q ≤ n in the revised manuscript. revision: yes

  2. Referee: [Uniqueness part] Uniqueness statement: uniqueness of smooth solutions is obtained only under the additional hypothesis that the density is sufficiently close to a constant in the Hölder norm. The paper should clarify whether this restriction is necessary for the C^0 estimates themselves or only for the uniqueness step, and how it affects the resolution of the Böröczky-Chen-Liu-Saroglou question.

    Authors: The C^0 estimates hold without symmetry assumptions and without the Hölder closeness condition on the density; this is what resolves the Böröczky-Chen-Liu-Saroglou question in the smooth category. The closeness condition is used only in the uniqueness result for smooth solutions. We will revise the abstract, introduction, and relevant statements to make this distinction explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; equivalence invoked as external fact

full rationale

The paper's chain establishes C^0 estimates for the L_p dual Minkowski problem (0<p<q≤n, no symmetry), proves uniqueness of smooth solutions when the density is Hölder-close to constant, and then invokes the equivalence 'uniqueness of the Minkowski type problem is equivalent to the validity of the Brunn-Minkowski inequality in a certain sense' to obtain the L_p Brunn-Minkowski inequality for dual quermassintegrals on origin-symmetric bodies. This equivalence is presented as a given external fact rather than derived from the paper's own estimates, fitted parameters, or self-citations. No self-definitional reduction, fitted-input-called-prediction, or load-bearing self-citation chain appears; the estimates and conditional uniqueness stand independently of the final inequality step. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only abstract available; no free parameters, invented entities, or additional axioms are visible beyond the stated equivalence and standard background in convex geometry.

axioms (2)
  • domain assumption Uniqueness of the Minkowski type problem is equivalent to the validity of the Brunn-Minkowski inequality in a certain sense
    Invoked explicitly to study the inequality for symmetric bodies
  • standard math Standard results from convex geometry and PDE theory on convex bodies
    Background assumed for the estimates and uniqueness proofs

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