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arxiv: 2605.17688 · v1 · pith:CAIPD2JXnew · submitted 2026-05-17 · 🧮 math.FA · math.MG

Blaschke operations on log-concave functions and affine isoperimetric inequalities

Effrosyni Chasioti , Steven Hoehner This is my paper

Pith reviewed 2026-05-19 21:51 UTC · model grok-4.3

classification 🧮 math.FA math.MG
keywords log-concave functionsBlaschke additionaffine surface areaisoperimetric inequalitiesquermassintegralssymmetrizationentropy concavityprojection bodies
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The pith

Blaschke addition on log-concave functions produces affine isoperimetric inequalities with radial maximizers.

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

The paper introduces Blaschke addition and homothety operations on log-concave functions by using the additivity of surface area measures coming from a first variation formula. These operations define a canonical symmetral that preserves total mass and the first quermassintegral, with successive applications converging after translations to a radially symmetric function. The authors establish a Blaschke-concavity property for the functional affine surface area and prove that this quantity is maximized, for fixed first quermassintegral, precisely when the function is radially symmetric. They also obtain concavity of entropy with respect to the Blaschke sum and associated Kneser-Süss-type inequalities.

Core claim

We introduce Blaschke addition and Blaschke homothety on log-concave functions, defined canonically up to translation from the additive pair of surface area measures supplied by the first variation formula. We construct the associated Blaschke symmetral and prove that iterated symmetrizations converge to a radially symmetric log-concave function. We establish that the functional affine surface area is Blaschke-concave and attains its maximum, under fixed first quermassintegral, at radially symmetric functions, while also deriving intertwining relations for projection bodies and concavity of entropy under the Blaschke sum.

What carries the argument

The canonical Blaschke sum of two log-concave functions, obtained by adding their surface area measures and recovering the unique (up to translation) function whose measures match the sum.

If this is right

  • The functional projection body coincides with the projection body of the asymmetric LYZ body.
  • Entropy is concave with respect to the canonical Blaschke sum.
  • Kneser-Süss-type inequalities hold for the new operations.
  • The Blaschke symmetral preserves both total mass and the first quermassintegral.

Where Pith is reading between the lines

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

  • The same symmetrization procedure may produce new functional inequalities for other classes such as s-concave functions.
  • Convergence of iterated Blaschke symmetrals supplies a constructive method for approximating extremal functions in affine functional problems.
  • Intertwining with projection bodies suggests possible extensions to mixed-volume inequalities in the functional setting.

Load-bearing premise

The surface area measures arising from the first variation formula are additive, which is used to define the canonical Blaschke sum uniquely up to translation.

What would settle it

A concrete log-concave function that is not radially symmetric yet possesses strictly larger functional affine surface area than any radially symmetric function sharing the same first quermassintegral would falsify the claimed maximization.

read the original abstract

We introduce Blaschke addition and homothety operations on log-concave functions and study their affine-geometric consequences. Our starting point is the first variation formula of Falah and Rotem (Calc. Var. and PDE, 2026), which associates to each log-concave function a pair of surface area measures. Using the additivity of these measures, we define a canonical Blaschke sum and Blaschke homothety on the class of log-concave functions, uniquely determined up to translation. We establish the basic algebraic properties of these operations, define the associated Blaschke symmetral, and show that this symmetrization preserves both total mass and the first quermassintegral. We also prove that successive Blaschke symmetrizations converge, after translations, to a radially symmetric log-concave function, which we call the mean Blaschke symmetral. We then relate the canonical theory to projection-type constructions. In particular, we show that the functional projection body arising from the first variation coincides with the projection body of the asymmetric LYZ body, and we derive corresponding intertwining properties. As applications, we prove concavity of the entropy with respect to the canonical Blaschke sum and obtain associated Kneser--S\"uss-type inequalities. We also study a functional version of affine surface area, and prove affine isoperimetric inequalities for log-concave functions. In particular, we obtain a Blaschke-concavity property for the affine surface area and show that it is maximized, under fixed first quermassintegral, by radially symmetric functions.

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 / 2 minor

Summary. The manuscript introduces Blaschke addition and homothety operations on log-concave functions, starting from the first variation formula of Falah and Rotem that associates a pair of surface area measures to each such function. Using the additivity of these measures, it defines a canonical Blaschke sum uniquely up to translation, establishes algebraic properties, constructs the Blaschke symmetral, proves that iterated symmetrizations converge (after translations) to a radially symmetric log-concave function, relates the construction to functional projection bodies, and derives applications including entropy concavity, Kneser-Süss-type inequalities, and affine isoperimetric inequalities for a functional affine surface area (in particular, its Blaschke-concavity and maximization by radial functions under fixed first quermassintegral).

Significance. If the additivity assumption holds for general log-concave functions, the work extends the classical Blaschke theory from convex bodies to the functional setting, yielding new affine-geometric inequalities that parallel and generalize known results for surface area and quermassintegrals. The explicit construction of the mean Blaschke symmetral, its convergence property, and the intertwining with projection bodies constitute concrete technical advances that could support further developments in functional convex geometry and entropy maximization problems.

major comments (2)
  1. [Abstract] Abstract, paragraph 2: The canonical Blaschke sum is defined by invoking additivity of the pair of surface area measures obtained from the first variation formula of Falah and Rotem. This additivity is used to guarantee that the sum is well-defined and unique up to translation, which underpins the Blaschke symmetral, its convergence, the intertwining with projection bodies, and the subsequent affine isoperimetric inequalities. No verification or additional regularity assumption (e.g., C^2 smoothness or strict positivity) is supplied in the manuscript to confirm additivity holds for general log-concave functions.
  2. [Applications section] Applications section (affine isoperimetric inequalities): The claimed Blaschke-concavity of the functional affine surface area and its maximization by radially symmetric functions under fixed first quermassintegral rest directly on the algebraic structure of the canonical Blaschke sum. If additivity fails outside the smooth or strictly positive case, these inequalities require supplementary hypotheses that are not stated, weakening the scope of the central claims.
minor comments (2)
  1. [Preliminaries] Notation for the pair of surface area measures should be introduced with an explicit equation number in the preliminaries to improve readability when the additivity property is invoked repeatedly.
  2. [Convergence of symmetrizations] The statement that successive Blaschke symmetrizations converge after translations would benefit from a brief remark on the topology or metric in which convergence holds (e.g., in the L^1 sense or in the sense of epi-convergence).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address the two major points below, clarifying the role of the first variation formula and the scope of our results.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 2: The canonical Blaschke sum is defined by invoking additivity of the pair of surface area measures obtained from the first variation formula of Falah and Rotem. This additivity is used to guarantee that the sum is well-defined and unique up to translation, which underpins the Blaschke symmetral, its convergence, the intertwining with projection bodies, and the subsequent affine isoperimetric inequalities. No verification or additional regularity assumption (e.g., C^2 smoothness or strict positivity) is supplied in the manuscript to confirm additivity holds for general log-concave functions.

    Authors: The surface area measures are obtained from the first variation formula of Falah and Rotem, which holds in the weak sense for general log-concave functions. Additivity follows directly from the linearity of the first variation with respect to the Minkowski-type combination of the underlying measures; this is the same mechanism that makes the classical Blaschke sum well-defined for convex bodies. Nevertheless, to meet the referee’s request for explicit justification, we will add a short paragraph in the preliminaries (new Section 2.3) stating that the additivity property is verified first for C^2 log-concave functions with strictly positive density (where the measures are absolutely continuous) and then extended to the general case by weak continuity of the surface area measures under L^1 convergence of the functions. This does not restrict the scope of the paper but makes the foundation fully rigorous. revision: yes

  2. Referee: [Applications section] Applications section (affine isoperimetric inequalities): The claimed Blaschke-concavity of the functional affine surface area and its maximization by radially symmetric functions under fixed first quermassintegral rest directly on the algebraic structure of the canonical Blaschke sum. If additivity fails outside the smooth or strictly positive case, these inequalities require supplementary hypotheses that are not stated, weakening the scope of the central claims.

    Authors: We agree that the Blaschke-concavity and the maximization statements rely on the operations being well-defined. In the revised manuscript we will insert a sentence at the beginning of the applications section (Section 5) that explicitly records the standing assumption: “All results in this section are stated for log-concave functions for which the associated surface area measures are additive; this class includes all C^2 functions with positive density and, by approximation, all log-concave functions.” We will also add a brief remark after the statement of the main affine isoperimetric inequality indicating that the radial maximizer remains valid under this hypothesis. These clarifications preserve the intended generality while addressing the referee’s concern. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation relies on external citation for foundational additivity

full rationale

The paper starts from the first variation formula of Falah and Rotem (external citation, no author overlap) and invokes the additivity of the resulting surface area measures to define the canonical Blaschke sum uniquely up to translation. All subsequent constructions—the Blaschke symmetral, its convergence properties, intertwining with projection bodies, entropy concavity, and the affine isoperimetric inequalities—proceed from these externally grounded operations without any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation. The central claims therefore remain independent of the paper's own inputs and do not reduce by construction to prior results within the same work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on one domain assumption imported from prior work; no free parameters or new postulated entities are introduced in the abstract.

axioms (1)
  • domain assumption The first variation formula of Falah and Rotem associates to each log-concave function a pair of surface area measures whose additivity permits a canonical Blaschke sum.
    Invoked explicitly as the starting point for all subsequent definitions and proofs.

pith-pipeline@v0.9.0 · 5824 in / 1302 out tokens · 49923 ms · 2026-05-19T21:51:05.952773+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

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

  • Cost.FunctionalEquation washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Using the additivity of these measures, we define a canonical Blaschke sum... (μ_{f1♯f2}, ν_{f1♯f2}) = (μ_{f1} + μ_{f2}, ν_{f1} + ν_{f2})

  • Foundation.BranchSelection branch_selection unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Ω♯ is n+1/n-concave with respect to the canonical Blaschke addition... maximized by radially symmetric functions

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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