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arxiv: 2606.25405 · v1 · pith:PI6HDRE3new · submitted 2026-06-24 · 🧮 math.AP · math.MG

The Lp centro-sectional Minkowski problem

Karoly J. Boroczky , Jiaqian Liu , Renqing You This is my paper

Pith reviewed 2026-06-25 21:07 UTC · model grok-4.3

classification 🧮 math.AP math.MG
keywords Lp Minkowski problemcentro-sectional measureBrunn-Minkowski inequalityconvex bodyaffine quermassintegraldual curvature measure
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The pith

The Lp centro-sectional Minkowski problem admits solutions for all p greater than 1 and q greater than 0.

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

The paper formulates an Lp version of the centro-sectional Minkowski problem by incorporating a real parameter q into the centro-sectional measure. It establishes existence of solutions in this setting, along with results on regularity and uniqueness of the resulting convex bodies. For sufficiently large p the work also derives Lp Brunn-Minkowski-type inequalities that extend classical volume inequalities to this affine context.

Core claim

The Lp centro-sectional Minkowski problem is solved by proving existence of convex bodies whose Lp centro-sectional measures equal a given data function, for every p>1 and q>0; the solutions are shown to be regular and unique under suitable conditions on the data, and Lp Brunn-Minkowski inequalities hold when p is large enough.

What carries the argument

The Lp centro-sectional measure, obtained by combining the Lp Minkowski combination with the centro-sectional measure of parameter q.

If this is right

  • Existence supplies a variational method for recovering convex bodies from their centro-sectional data in the Lp regime.
  • Regularity results imply that smooth data produce smooth solutions when p>1.
  • Uniqueness statements allow the problem to serve as a characterization tool for convex bodies.
  • Lp Brunn-Minkowski inequalities for large p give quantitative control on how volume behaves under Lp combinations of centro-sectional measures.

Where Pith is reading between the lines

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

  • The same existence technique may adapt to other dual curvature measures that carry an extra real parameter.
  • The inequalities could be used to bound affine surface areas in optimization problems over convex sets.
  • Testing the boundary case p=1 might reveal whether the large-p restriction is essential or merely technical.

Load-bearing premise

The centro-sectional measures introduced with real parameter q are well-defined and can be extended compatibly to the Lp setting.

What would settle it

An explicit pair p>1, q>0 and a positive continuous function on the sphere for which no convex body exists whose Lp centro-sectional measure equals that function.

read the original abstract

As part of Lutwak's broadening of the Brunn-Minkowski theory, and extending the notion of affine quermassintegrals and dual curvature measure discussed by Milman, Yehudayoff and Huang, Lutwak, Yang and Zhang, centro-sectional measures with real parameter q have been recently introduced by Cai, Leng, Wu, Xi. In this paper, we introduce the Lp cross sectional Minkowski problem analogously to the Lp dual Minkowski problem formulated by Lutwak, Yang and Zhang. We solve the Lp dual Minkowski problem for p>1 and q>0, discuss the regularity and uniqueness of the solution, and prove Lp Brunn-Minkowski-type inequalities when $p$ is relatively large.

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

1 major / 1 minor

Summary. The paper introduces the Lp centro-sectional Minkowski problem by extending the centro-sectional measures with real parameter q (recently introduced by Cai, Leng, Wu, Xi) to the Lp setting, in analogy with the Lp dual Minkowski problem. It claims to solve the problem for p>1 and q>0, discuss regularity and uniqueness of solutions, and prove Lp Brunn-Minkowski-type inequalities for relatively large p.

Significance. If the central claims hold, the work would extend the affine Brunn-Minkowski theory by incorporating Lp surface area measures into the centro-sectional framework, yielding new existence results, regularity statements, and inequalities that build on prior affine quermassintegrals and dual curvature measures.

major comments (1)
  1. [Introduction] The existence, regularity, and uniqueness statements for p>1, q>0 rest on the centro-sectional measures (with parameter q) being well-defined, positive, and satisfying the necessary variational or PDE properties when combined with the Lp surface area measure. The manuscript cites the recent introduction by Cai et al. but supplies no independent verification or re-derivation of these properties in the Lp context (Introduction and the problem formulation section).
minor comments (1)
  1. The abstract refers to the 'Lp cross sectional Minkowski problem' while the title uses 'centro-sectional'; ensure consistent terminology is used throughout the manuscript.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the recommendation for major revision. We address the single major comment point by point below.

read point-by-point responses

  1. Referee: [Introduction] The existence, regularity, and uniqueness statements for p>1, q>0 rest on the centro-sectional measures (with parameter q) being well-defined, positive, and satisfying the necessary variational or PDE properties when combined with the Lp surface area measure. The manuscript cites the recent introduction by Cai et al. but supplies no independent verification or re-derivation of these properties in the Lp context (Introduction and the problem formulation section).

    Authors: The centro-sectional measures with parameter q and their core properties (well-definedness, positivity, and variational formulas) are established in Cai-Leng-Wu-Xi and are purely geometric, independent of the Lp parameter. The Lp centro-sectional Minkowski problem is obtained by replacing the classical surface area measure with its Lp counterpart in the standard manner, as done for the Lp dual Minkowski problem. The existence, regularity, and uniqueness proofs adapt the variational and PDE techniques from that setting using the cited properties directly. We agree that an explicit bridge between the two frameworks would strengthen the presentation. We will therefore insert a short paragraph after the problem formulation that recalls the relevant statements from Cai et al. and notes their immediate applicability when the measure is paired with the Lp surface area measure. revision: yes

Circularity Check

0 steps flagged

No circularity; new problem solved via external prior framework

full rationale

The paper defines and solves the Lp centro-sectional Minkowski problem by direct analogy to the Lp dual Minkowski problem, citing Cai-Leng-Wu-Xi (distinct authors) for the underlying centro-sectional measures with parameter q. No load-bearing step reduces by construction to a self-defined quantity, fitted input, or self-citation chain; the existence/uniqueness claims rest on standard convex-geometric techniques applied to the externally introduced measures. This is a normal non-circular extension of prior literature.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted from the full manuscript.

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