Convexity of Radial Mean Bodies via an extension of Ball's Bodies

Dylan Langharst

arxiv: 2603.14134 · v3 · submitted 2026-03-14 · 🧮 math.MG · math.FA

Convexity of Radial Mean Bodies via an extension of Ball's Bodies

Dylan Langharst This is my paper

Pith reviewed 2026-05-15 10:59 UTC · model grok-4.3

classification 🧮 math.MG math.FA

keywords convex geometryradial mean bodieslog-concave functionsPrékopa theoremconvexityintegral inequalitieshomogeneous functionsBall bodies

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The pith

Radial mean bodies of convex sets are convex for every exponent p greater than -1.

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

The paper extends Keith Ball's classical result on integrals of log-concave functions along rays, weighted by r to the power p-1, into the range p between -1 and 0. This extension is applied to the radial p-th mean bodies R_p K introduced by Gardner and Zhang, establishing their convexity in the regime that had remained open since 1998. A reader would care because the result unifies the convexity statement across all p greater than -1 and supplies a new proof that also covers the original positive-p case. The argument reduces the n-dimensional question to a two-dimensional inequality obtained from Prékopa's theorem on log-concave functions.

Core claim

If g is an integrable log-concave function on R^n that attains its maximum at the origin, then the map sending x to (p/g(0) times the integral from 0 to infinity of r^{p-1}(g(r x) - g(0)) dr) raised to the power -1/p is a positively 1-homogeneous convex function for every p in (-1, 0). The same statement holds for p greater than 0 by the earlier work of Ball. As a direct consequence, the radial p-th mean body R_p K of any convex body K is convex for all p greater than -1.

What carries the argument

The p-weighted radial integral transform of a log-concave function g, shown to be convex by reduction of the defining inequality to a two-dimensional case derived from Prékopa's theorem.

If this is right

Convexity of R_p K now holds uniformly for every p greater than -1, closing the gap left open by Gardner and Zhang.
The new inequality supplies a single proof covering both the positive-p regime and the negative-p regime.
Any geometric inequality previously restricted to p greater than or equal to zero can now be considered for p in (-1,0) as well.
The reduction technique itself may be reusable for other integral representations involving log-concave densities.

Where Pith is reading between the lines

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

The same convexity statement might extend to certain non-log-concave functions if a suitable replacement for Prékopa's theorem can be found.
Volume or surface-area functionals built from R_p K could inherit monotonicity or inequality properties in the newly accessible negative range.
The two-dimensional reduction may connect to other Brunn-Minkowski-type inequalities in low dimensions that have not yet been exploited for radial means.

Load-bearing premise

The n-dimensional integral inequality for the extended Ball body reduces to a two-dimensional inequality from Prékopa's theorem precisely when g is integrable, log-concave, and attains its maximum at the origin.

What would settle it

An explicit convex body K together with a value p in (-1,0) for which the radial mean body R_p K fails to be convex would disprove the central claim.

read the original abstract

In this work, we extend a classical theorem of Keith Ball on integrals of log-concave functions along rays against the weight $r^{p-1}$ to the previously inaccessible regime $p\in (-1,0)$: if $g:\mathbb R^n\to\mathbb R_+$ is an integrable log-concave function which attains its maximum at the origin, then \[ x\mapsto \left(\frac{p}{g(o)}\int_{0}^{\infty}r^{p-1}(g(rx)-g(o))\mathrm{d}\,r\right)^{-\frac{1}{p}} \] is a positively 1-homogeneous convex function on $\mathbb{R}^n$. Our approach also provides a new proof of the original regime $p> 0$. The argument is based on a reduction to a two-dimensional inequality derived from Pr\'ekopa's theorem, which may be of independent interest. As a consequence of this extension, we resolve a nearly 30-year-old question of Richard Gardner and Gaoyong Zhang in the affirmative. In 1998, R. Gardner and G. Zhang introduced the radial $p$th mean bodies $R_p K$ of a convex body $K\subset \mathbb{R}^n$ for $p>-1$. Furthermore, they established that $R_p K$ is convex for $p\geq 0$, but the convexity of $R_p K$ for $p\in (-1,0)$ remained open. We prove that $R_p K$ is convex for all $p>-1$.

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.

Desk Editor's Note private letter to a colleague

Extends Ball's theorem to p in (-1,0) and settles convexity of R_p K for all p>-1, with the 2D Prékopa reduction as the key step that needs verification for the singularity.

read the letter

This paper proves that R_p K is convex for p in (-1,0), which closes the open question Gardner and Zhang left in 1998. It does this by extending Ball's theorem on the convexity of the functional built from integrals of log-concave functions against r^{p-1} to the negative range, and it also gives a fresh proof for p>0. The strategy reduces the n-dimensional claim to a two-dimensional inequality derived from Prékopa's theorem on marginals, which is the part that looks new and potentially reusable. Once that extension holds, the convexity of the radial mean bodies follows directly without extra work. The argument stays within established tools and avoids circularity. The main soft spot is the singularity at the origin when p is negative. The weight r^{p-1} is no longer locally integrable in the usual sense, and the sign change in the integrand has to be offset exactly by the factor p to preserve convexity and 1-homogeneity. The paper states that the resulting 2D inequality still works when g is integrable, log-concave, and maximized at zero, but the stress-test concern is fair: without an explicit check or independent computation in the 2D case, it is not obvious that the inequality survives uniformly in all directions. If that step is solid, the result is fine; if there is a gap, the extension to negative p remains open. This is for people working in convex geometry who already know Ball's theorem and the Gardner-Zhang bodies. A reader who follows integral inequalities or Brunn-Minkowski variants will get the most from it. The math and citations look standard and grounded. I would bring it to a reading group to walk through the 2D reduction. I would cite it for the convexity statement if I needed it. It deserves a serious referee because the open problem is real and the proposed fix is concrete.

Referee Report

1 major / 1 minor

Summary. The paper extends Keith Ball's theorem on integrals of log-concave functions weighted by r^{p-1} to the regime p ∈ (-1,0). For an integrable log-concave g: R^n → R_+ attaining its maximum at the origin, the map x ↦ [p/g(0) ∫_0^∞ r^{p-1} (g(rx) - g(0)) dr ]^{-1/p} is shown to be positively 1-homogeneous and convex. The proof reduces the n-dimensional statement to a two-dimensional inequality obtained from Prékopa's theorem. As a consequence, the radial p-th mean bodies R_p K of a convex body K are convex for all p > -1, resolving the open question posed by Gardner and Zhang in 1998. A new proof is also given for the original case p > 0.

Significance. If the central reduction holds, the result resolves a nearly 30-year-old open problem in convex geometry by establishing convexity of R_p K for negative p. It also supplies a new proof of Ball's theorem for p > 0 and isolates a two-dimensional inequality that may be of independent interest for log-concave functions and marginals.

major comments (1)

[the reduction to the two-dimensional inequality] The reduction of the n-dimensional convexity claim to the two-dimensional inequality derived from Prékopa's theorem (described in the abstract and the proof strategy) must be verified explicitly for p ∈ (-1,0). The weight r^{p-1} introduces a non-integrable singularity at the origin together with a sign change; it is not immediate that the resulting 2D inequality still yields a convex, positively 1-homogeneous function uniformly in all directions.

minor comments (1)

[statement of the main theorem] The abstract states that g is integrable and log-concave with maximum at the origin; the manuscript should clarify whether the integrability assumption is used only for the existence of the integral or also for the application of Prékopa's theorem on marginals.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the significance of extending Ball's theorem to p ∈ (-1,0). We address the major comment below.

read point-by-point responses

Referee: [the reduction to the two-dimensional inequality] The reduction of the n-dimensional convexity claim to the two-dimensional inequality derived from Prékopa's theorem (described in the abstract and the proof strategy) must be verified explicitly for p ∈ (-1,0). The weight r^{p-1} introduces a non-integrable singularity at the origin together with a sign change; it is not immediate that the resulting 2D inequality still yields a convex, positively 1-homogeneous function uniformly in all directions.

Authors: We agree that explicit verification is needed for p ∈ (-1,0). In the manuscript (Section 3), the reduction proceeds by applying Prékopa's theorem to obtain a 2D inequality for the weighted integral along rays, which is then used to establish convexity of the n-dimensional map. For p ∈ (-1,0), although r^{p-1} is singular at the origin, the factor (g(rx) - g(0)) vanishes at r = 0. Log-concavity of g implies that g(rx) - g(0) = O(r) near r = 0 (in fact, the difference is controlled by the concavity of log g along the ray), which ensures integrability of the product since the exponent p-1 > -2 and the linear vanishing compensates the singularity. We split the integral at r=1 and handle the near-zero part by a direct estimate using the supporting hyperplane property of log-concave functions. The resulting 2D expression is shown to be convex and positively 1-homogeneous by the same monotonicity and convexity preservation arguments as for p > 0, with the sign of p accounted for by factoring out the negative constant (which does not affect convexity). The homogeneity holds uniformly in all directions by construction. We will add a dedicated lemma and expanded estimates in the revision to make this verification fully explicit, including the convergence proof and a numerical check for an exponential example. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses external Prékopa theorem for independent 2D reduction

full rationale

The paper's central result extends Ball's theorem on integrals of log-concave functions to p in (-1,0) by reducing the n-dimensional statement to a two-dimensional inequality obtained from Prékopa's theorem on marginals. This reduction is presented as a new argument that also reproves the p>0 case, with the convexity of R_p K for p>-1 following directly as a consequence. No equation or step equates the target convexity statement to a fitted parameter, a self-defined quantity, or a load-bearing self-citation whose own justification is internal to the paper. Prékopa's theorem is invoked as an independent, externally established tool, satisfying the criteria for a self-contained derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard background results in convex analysis without introducing new free parameters or postulated entities.

axioms (2)

standard math Prékopa's theorem on preservation of log-concavity under marginals
Invoked to obtain the two-dimensional inequality that reduces the original problem.
domain assumption Log-concavity and integrability of g with maximum at the origin
Stated as the hypothesis under which the extended integral expression is convex.

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discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

x↦(p/g(o)∫_0^∞ r^{p-1}(g(rx)-g(o))dr)^{-1/p} is positively 1-homogeneous convex (Theorem 1, p∈(-1,0))
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

reduction to 2D inequality (1) via Prékopa marginals on f(r,s)=g(ru+sθ)

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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.
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