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arxiv: 2407.20064 · v4 · submitted 2024-07-29 · 🧮 math.AP · math.FA· math.MG

The Weighted L^p Minkowski Problem

Dylan Langharst , Jiaqian Liu , Shengyu Tang This is my paper

Pith reviewed 2026-05-23 23:00 UTC · model grok-4.3

classification 🧮 math.AP math.FAmath.MG
keywords Minkowski problemL^p Minkowski problemweighted surface area measuresconvex bodiesexistenceuniquenessrotationally invariant measuresdegree theory
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The pith

The weighted L^p Minkowski problem admits solutions for all real p when the measure is rotationally invariant.

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

The paper tries to establish existence of convex bodies that realize any given rotationally invariant measure on the sphere as their weighted L^p surface area measure, for every real number p. Symmetry conditions on the measure are imposed in some cases to secure the existence proof. Uniqueness of the body holds when p is at least 1 and the measure satisfies an additional concavity condition. Results for measures of small total mass are obtained separately by topological degree arguments. A reader would care because the results supply a uniform existence theory that covers a wider family of weights than the Gaussian case alone.

Core claim

For rotationally invariant measures we prove existence of a convex body whose weighted L^p surface area measure equals the given measure for every real p, with symmetry assumptions used in certain instances. Uniqueness holds for p greater than or equal to 1 under a concavity assumption. In the small-mass regime existence follows from degree theory.

What carries the argument

The weighted L^p surface area measure of a convex body with respect to a rotationally invariant weight function.

If this is right

  • Existence holds for all real p with symmetry assumptions in some cases.
  • Uniqueness holds for p at least 1 under a concavity assumption.
  • Existence in the small-mass regime follows from degree theory.

Where Pith is reading between the lines

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

  • The same symmetry-based approach could be tested on other weighted problems in convex geometry.
  • Relaxing rotational invariance would require new techniques and might produce different existence thresholds.
  • The degree-theory argument for small mass may transfer to related Minkowski-type problems with weights.

Load-bearing premise

The given measures must be rotationally invariant.

What would settle it

A concrete rotationally invariant measure on the sphere for which no convex body has the corresponding weighted L^p surface area measure.

read the original abstract

The Minkowski problem in convex geometry concerns showing that a given Borel measure on the unit sphere is, up to perhaps a constant, some type of surface area measure of a convex body. Two types of Minkowski problems in particular are an active area of research: $L^p$ Minkowski problems, introduced by Lutwak and (Lutwak, Yang, and Zhang), and weighted Minkowski problems, introduced by Livshyts. For the latter, the Gaussian Minkowski problem, whose primary investigators were (Huang, Xi and Zhao), is the most prevalent. In this work, we consider weighted surface area in the $L^p$ setting. We propose a framework going beyond the Gaussian setting by focusing on rotationally invariant measures, mirroring the recent development of the Gardner-Zvavitch inequality for rotationally invariant, log-concave measures. Our results include existence for all $p \in \mathbb R$ (with symmetry assumptions in certain instances). We also have uniqueness for $p \geq 1$ under a concavity assumption. Finally, we obtain results in the so-called "small mass regime" using degree theory, as instigated in the Gaussian case by (Huang, Xi and Zhao).

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 develops a framework for the weighted L^p Minkowski problem restricted to rotationally invariant measures. It claims existence of convex bodies realizing given data for all real p (with additional symmetry assumptions in some cases), uniqueness for p ≥ 1 under an extra concavity hypothesis on the measure, and existence results in the small-mass regime obtained via topological degree theory, extending the Gaussian setting of Huang-Xi-Zhao.

Significance. If the arguments hold, the work supplies a natural extension of the Gaussian Minkowski problem to the larger class of rotationally invariant log-concave measures, paralleling recent progress on the Gardner-Zvavitch inequality. The explicit use of degree theory in the small-mass regime is a methodological strength that could be reusable in related problems.

minor comments (2)
  1. [Abstract] Abstract: the parenthetical remark on symmetry assumptions is vague; a single sentence listing the precise rotational-invariance hypotheses that appear in each existence theorem would improve readability.
  2. The manuscript should include a short comparison table or paragraph contrasting the new results with the Gaussian case of Huang-Xi-Zhao and with the unweighted L^p theory of Lutwak-Yang-Zhang.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report, so there are no individual points requiring response or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity; results rely on degree theory and standard tools under explicit symmetry assumptions

full rationale

The paper proves existence for all real p (with symmetry) and uniqueness for p ≥ 1 (with concavity) for the weighted L^p Minkowski problem on rotationally invariant measures, plus small-mass results via degree theory. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the framework explicitly conditions results on rotational invariance and log-concavity from the outset, mirroring but not deriving from prior work. Degree theory and convex-geometry tools provide independent content. No equations or claims in the abstract or described methods exhibit the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard background results from convex geometry and nonlinear analysis; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)
  • standard math Existence and basic properties of convex bodies and their surface area measures in Euclidean space.
    Invoked throughout the setup of the Minkowski problem.

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