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arxiv: 2509.04108 · v2 · submitted 2025-09-04 · 🧮 math.FA · math.MG· math.PR

On stochastic forms of functional isoperimetric inequalities

Francisco Mar\'in Sola This is my paper

Pith reviewed 2026-05-18 19:18 UTC · model grok-4.3

classification 🧮 math.FA math.MGmath.PR
keywords stochastic isoperimetric inequalitiesp-concave functionsfunctional quermassintegralsZhang affine Sobolev inequalityrandom modelsconvex measuresfunctional inequalities
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The pith

Zhang's affine Sobolev inequality holds in expectation for random models of p-concave functions.

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

The paper gives a probabilistic reading of functional isoperimetric inequalities inside the class of p-concave functions by using random models already studied in the literature. It first proves a stochastic isoperimetric inequality for a functional version of the classical quermassintegrals; this immediately produces a Sobolev-type inequality in the random setting. The central result is that Zhang's affine Sobolev inequality remains valid after taking expectations over these random p-concave functions. The same arguments recover both the usual geometric inequalities and the deterministic functional versions, and they yield a new generalization of Zhang's inequality that applies only to p-concave functions under convex measures.

Core claim

By working with random models of p-concave functions, a stochastic isoperimetric inequality for functional quermassintegrals is established; this yields a Sobolev-type inequality as a special case. Zhang's affine Sobolev inequality then holds after taking expectation with respect to the random model. The results recover the geometric and deterministic statements, and they produce a generalization of Zhang's inequality restricted to p-concave functions in the setting of convex measures.

What carries the argument

Random models of p-concave functions, used to define expectations of functional quermassintegrals that obey stochastic versions of the classical isoperimetric inequalities.

If this is right

  • A stochastic isoperimetric inequality holds for the functional extension of quermassintegrals.
  • A Sobolev-type inequality follows directly as a special case in the random setting.
  • Zhang's affine Sobolev inequality is valid after taking expectation over the random model.
  • Both the geometric analogues and the ordinary deterministic inequalities are recovered as limiting cases.
  • Zhang's inequality extends to p-concave functions when the underlying measure is convex.

Where Pith is reading between the lines

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

  • Similar stochastic extensions could be attempted for other functional inequalities that involve quermassintegrals or affine invariants.
  • The approach opens the possibility of proving inequalities by averaging over random samples rather than working directly with worst-case functions.
  • Numerical sampling of the random models might provide a practical way to test or approximate the constants in these inequalities.
  • The results suggest that convexity assumptions on the measure can be relaxed while still preserving the affine Sobolev bound inside the p-concave class.

Load-bearing premise

The random models of p-concave functions are regular enough that the functional quermassintegrals and their expectations are well-defined and satisfy the needed measurability and integrability conditions.

What would settle it

Construct an explicit random model of p-concave functions, compute the expectation of the affine Sobolev functional, and check whether the resulting value satisfies the claimed bound; a single counterexample where the expectation lies strictly outside the deterministic bound would disprove the claim.

Figures

Figures reproduced from arXiv: 2509.04108 by Francisco Mar\'in Sola.

Figure 1
Figure 1. Figure 1: Comparative of two stochastic models of a bidi￾mensional centered gaussian Our first main result reads as follows [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original abstract

We present a probabilistic interpretation of several functional isoperimetric inequalities within the class of $p$-concave functions, building on random models for such functions introduced by P. Pivovarov and J. Rebollo-Bueno. First, we establish a stochastic isoperimetric inequality for a functional extension of the classical quermassintegrals, which yields a Sobolev-type inequality in this random setting as a particular case. Motivated by the latter, we further show that Zhang's affine Sobolev inequality holds in expectation when dealing with these random models of $p$-concave functions. Finally, we confirm that our results recover both their geometric analogues and deterministic counterparts. As a consequence of the latter, we establish a generalization of Zhang's affine Sobolev inequality restricted to $p$-concave functions in the context of convex measures.

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 develops stochastic versions of functional isoperimetric inequalities for p-concave functions by leveraging random models introduced by Pivovarov and Rebollo-Bueno. It first proves a stochastic isoperimetric inequality involving functional extensions of quermassintegrals, from which a Sobolev-type inequality follows in the random setting. Building on this, the authors show that Zhang's affine Sobolev inequality holds when expectations are taken over these random models. The work recovers the corresponding geometric inequalities and deterministic functional inequalities as special cases, and concludes with a generalization of Zhang's inequality to the setting of convex measures restricted to p-concave functions.

Significance. If the central claims are rigorously established, the manuscript provides a useful probabilistic bridge between stochastic geometry and functional inequalities, extending classical results to random p-concave models while recovering known deterministic and geometric statements. The explicit recovery of prior results and the extension to convex measures constitute verifiable strengths that would strengthen the contribution if the integrability and measurability steps are completed.

major comments (1)
  1. [Section on expectation form of Zhang's inequality (following the stochastic Sobolev derivation)] The derivation that Zhang's affine Sobolev inequality passes to the expectation (the central claim) relies on the functional quermassintegrals being integrable with respect to the Pivovarov-Rebollo-Bueno random model. No tail estimates, domination arguments, or measurability verification for these random variables appear to be supplied, which is required before the inequality E[LHS] ≤ E[RHS] can be asserted.
minor comments (1)
  1. [Introduction and setup of random models] Clarify the precise function space (Sobolev or Orlicz) in which the random p-concave functions are assumed to lie almost surely, to make the functional quermassintegrals well-defined.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. The major comment identifies a technical point that requires clarification and strengthening in the manuscript. We address it point by point below and will incorporate the necessary revisions.

read point-by-point responses

  1. Referee: [Section on expectation form of Zhang's inequality (following the stochastic Sobolev derivation)] The derivation that Zhang's affine Sobolev inequality passes to the expectation (the central claim) relies on the functional quermassintegrals being integrable with respect to the Pivovarov-Rebollo-Bueno random model. No tail estimates, domination arguments, or measurability verification for these random variables appear to be supplied, which is required before the inequality E[LHS] ≤ E[RHS] can be asserted.

    Authors: We agree that explicit verification of integrability and measurability is required to justify passing the inequality to the expectation. In the revised version we will insert a new technical subsection immediately following the stochastic Sobolev derivation. This subsection will contain: (i) a proof that the functional quermassintegrals are measurable random variables on the underlying probability space, using the continuity of the quermassintegral functionals with respect to the appropriate topology on p-concave functions and the measurability of the Pivovarov-Rebollo-Bueno construction; (ii) tail estimates obtained from the concentration properties of the random model for p-concave functions; and (iii) a domination argument showing that the relevant random variables are bounded above by an integrable random variable whose expectation is finite and independent of the particular realization. These additions will allow us to invoke the dominated convergence theorem (or monotone convergence where applicable) and thereby rigorously justify E[LHS] ≤ E[RHS]. The same estimates will also clarify the passage to the deterministic limit. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations extend external random models to stochastic inequalities without internal reduction

full rationale

The paper cites the Pivovarov-Rebollo-Bueno random models for p-concave functions as an external foundation and derives stochastic isoperimetric inequalities, a Sobolev-type result in expectation, and recovery of deterministic Zhang affine Sobolev and geometric analogues. These steps use the cited models' definitions for quermassintegrals and expectations rather than defining the target quantities in terms of themselves or fitting parameters inside the paper. No self-citation load-bearing, ansatz smuggling, or uniqueness theorems from the present authors appear; the chain remains self-contained against the external benchmarks and does not reduce any claimed prediction to an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the existence and regularity properties of the random models for p-concave functions from prior work; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The random models of p-concave functions are measurable and integrable with respect to the relevant measures so that expectations of functional quermassintegrals exist.
    Invoked to state the stochastic inequalities and their expectation forms.

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