On a generalization of the Hermite-Hadamard inequality and applications in convex geometry

Bernardo Gonz\'alez Merino

REVIEW 3 major objections 4 minor 1 cited by

On a generalization of the Hermite-Hadamard inequality and applications in convex geometry

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves a generalized Hermite-Hadamard inequality for convex functions composed with concave functions on 0-symmetric convex bodies, and uses it to determine the optimal constant in a volume-versus-central-section bound with full…

desk verdict New inequality with a clean proof and the optimal constant for Santos's question; the equality characterisation is false for constant functions and needs a small fix. read the letter →

arxiv 1908.06426 v3 pith:TTKOC3SD submitted 2019-08-18 math.FA math.MG

Bernardo González Merino This is my paper · ORCID

classification math.FAmath.MG MSC 52A2052A3852A40

keywords Hermite-Hadamardinequalityconvexgeometry0-symmetricbodiesSchwarzsymmetrizationBrunn-Minkowskilog-concavefunctionsvolumeofcentralsectionsoptimalconstants

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a sharp, higher-dimensional extension of the Hermite-Hadamard inequality. For any 0-symmetric convex body $C$, any concave nonnegative function $f$ on it, and any convex nondecreasing $\varphi$ with $\varphi(0)=0$, the average of $\varphi(f(x))$ over $C$ is no larger than the one-dimensional average of $\varphi(f(0)(1+t))$ over $t\in[-1,1]$. When $\varphi$ is strictly convex, equality forces $C$ to be a generalized cylinder and $f$ to be affine and vanish on one base. The inequality yields an optimal constant $c_n=2^n/n$ for bounding the volume of a convex body by the volume of a central hyperplane section when its projection on the normal line is symmetric, and it gives a sharp Hermite-Hadamard bound for log-concave functions. The proof runs through Schwarz symmetrization and a volume-preserving comparison with a cylinder, so the same mechanism carries both the inequality and its equality cases.

What carries the argument

The argument is carried by a one-dimensional comparison lemma for convex functions: for $\varphi$ convex, nondecreasing, with $\varphi(0)=0$, the symmetric sum $\varphi(a-r)+\varphi(a+r)$ is no larger than $\varphi(a-\gamma r)+\varphi(a+\gamma r)$ whenever $a\ge \gamma r\ge0$ and $\gamma\ge1$ (inequality (9)). This is applied after symmetrizing $C$ with respect to the direction $e_1$ where an affine majorant $g$ of $f$ has its steepest descent. The symmetrized body $C'$ is sandwiched between two cylinders, $R_{t_0}\subset C'\subset R_0$, built from the section $M_0'$ and the extreme section $M_{t_0}'$; a volume-preserving intermediate cylinder $R$ with the same volume as $C'$ is selected. The slicewise comparison then shifts the $\delta$-slope in $g(0)+(t/t_0)\delta$ up to the full slope $g(0)(1+t/t_0)$, turning the integral over $C$ into the one-dimensional integral on the right. The equality case is decided by Brunn-Minkowski: equal section volumes force all sections of $C$ to be translates of one $(n-1)$-dimensional body, which is exactly a generalized cylinder.

What would settle it

Take any 0-symmetric convex body $C$ in dimension $n\ge2$ that is not a generalized cylinder, for example a Euclidean ball, set $f(x)=0$ for all $x$, and set $\varphi(t)=t^2$, which is strictly convex with $\varphi(0)=0$. Then the left and right sides of Theorem 1.2 are both 0, so equality holds, even though the equality characterization would require $C$ to be a generalized cylinder. Checking this example directly would settle that the 'only if' part of the characterization is false without an extra nonconstant assumption on $f$.

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Extended reading notes

Core claim

The central claim is Theorem 1.2: if $C$ is an $n$-dimensional 0-symmetric compact convex set, $f:C\to[0,\infty)$ is concave, and $\varphi:[0,\infty)\to[0,\infty)$ is convex, not identically zero, with $\varphi(0)=0$, then $$\frac{1}{|C|}\int_C \varphi(f(x))\,dx \le \frac12\int_{-1}^{1}\varphi(f(0)(1+t))\,dt.$$ When $\varphi$ is strictly convex, equality occurs exactly when, up to rotation, $C=[-x_0,x_0]+\{0\}\times C_0$ for some $C_0\in\mathcal{K}^{n-1}_0$ and $f$ is affine with $f(-x_0+x)=0$ for every $x$ in the opposite base. From this the paper derives a volumetric inequality: if $K$ is convex and $H$ is a subspace whose projection $P_H K$ is symmetric, then $|K|\le \frac{2^{n-i}}{n-i+1}|P_H K||K\cap H^\perp|$, with equality characterized by a generalized cylinder structure. The same functional inequality gives a Hermite-Hadamard estimate for log-concave functions in terms of $\log(f(0)/f_{\min})$. The proof reduces the problem to one dimension by orienting along a support line, replacing the body by its Schwarz symmetrization, and sandwiching that symmetric body between two cylinders whose slices are concentric balls.

Load-bearing premise

The proof of the equality case assumes the supporting line used to bound f actually tilts along the chosen direction; when f is constant, the key comparison step divides by this tilt and the equality characterization as stated misses those cases.

Editorial extensions

If this is right

For $\varphi(t)=t^\alpha$, $\alpha\ge1$, Corollary 2.2 gives $\frac{1}{|C|}\int_C f^\alpha \le \frac{2^\alpha}{\alpha+1} f(0)^\alpha$, with equality if and only if $C$ is a generalized cylinder and $f$ is affine and vanishes on one base.
The volume inequality with the optimal constant $2^n/n$ answers the 2017 question: when a convex body's projection on a line is $[-e_1,e_1]$, the body's volume is at most $(2^n/n)$ times the volume of its intersection with the hyperplane $e_1^\perp$; equality cases are explicitly described.
For log-concave $f$, Theorem 1.3 gives $\frac{1}{|C|}\int_C f \le f_{\min}\frac{(f(0)/f_{\min})^2-1}{\log((f(0)/f_{\min})^2)}$, with equality only for generalized cylinders and log-affine functions attaining the minimum on one base.
Theorem 3.2 removes the symmetry assumption in the hyperplane case: for any $K\in\mathcal{K}^n$ and $H\in\mathcal{L}^n_{n-1}$, $|K|\le |P_H K||K\cap(x_{P_H K}+H^\perp)|$, with equality characterized. This choice can improve on the earlier centroid-section bound of (2).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One extension not pursued in the paper: because the core comparison only needs convexity, monotonicity, and $\varphi(0)=0$, the same proof should yield exponential-moment bounds $\frac{1}{|C|}\int_C e^{\lambda f(x)}dx \le \frac12\int_{-1}^1 e^{\lambda f(0)(1+t)}dt$ for $\lambda>0$, giving Laplacian-type estimates for concave functions on symmetric bodies.
The sandwich-by-cylinders argument is robust enough to suggest that the optimal constant extends to higher moments of section volumes: for any even moment of $|K\cap(x+H^\perp)|$, the extremal body should be the same generalized cylinder.
A natural testable extension is to non-symmetric bodies by replacing the role of 0 with the centroid or a related center; the inequality would then compare the average of $\varphi(f)$ to a one-dimensional integral centred at that point, possibly with a dimension-dependent constant.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The main inequality is genuinely new and the application is worth having. The paper answers Santos's question with the optimal constant, and the Hermite-Hadamard-type bound with the center fixed at the origin is a real extension of the Milman–Pajor and Borell versions. The proof idea—compare the original body to a Schwarz-symmetrized cylinder with the same slices, then use convexity to push the integral to a two-point boundary term—is clean and, as far as I can tell, correct for the inequality itself.

The soft spot is the equality characterisation, and it is real but local. In Theorem 1.2, if f is identically zero then both sides are zero for every 0-symmetric C, so the “only if” part cannot require the generalized cylinder structure. The proof breaks at the point where γ = g(0)/δ is introduced: δ = 0 is not handled, and δ = 0 is exactly the constant-zero case. The fix is straightforward: either exclude f ≡ 0 from the equality statement, or add a sentence that if f ≡ 0 equality is trivial and otherwise f(0) > 0 and one can choose the supporting affine function with δ > 0. Theorem 1.3 inherits the same issue: constant positive log-concave functions give equality for every C, so the equality case there is false as stated. That is also easily fixed by excluding constant functions or stating the degenerate case separately.

There is also a sign typo in Remark 2.1: f(x) = ⟨x, e1⟩ − 1 is nonpositive on the cylinder, so it should be f(x) = 1 − ⟨x, e1⟩. Minor, but it should be corrected.

The citation pattern looks fine; the paper builds on standard results and does not fit parameters. The application to the volume question is solid because the critical part of Theorem 1.1 only needs the inequality, not the flawed equality case.

Bottom line: the main inequality and the constant for Santos's question are correct and worth publishing. The equality statements need revision before the paper is final. I would send it to a serious referee; a competent referee will find these issues quickly, but they are fixable and do not undermine the main results.

Referee Report

3 major / 4 minor

Summary. The paper proves a functional inequality of Orlicz-Hermite-Hadamard type: for an n-dimensional 0-symmetric convex body C, a concave function f:C->[0,∞), and a convex function φ:[0,∞)->[0,∞) with φ(0)=0, one has (1/|C|)∫_C φ(f(x))dx ≤ (1/2)∫_{-1}^{1} φ(f(0)(1+t))dt. It also states equality characterizations for strictly convex φ, derives a corresponding estimate for log-concave functions (Theorem 1.3), and applies the power case to obtain the optimal constant in Francisco Santos's question on bounding |K| by |P_H K||K∩H^⊥| (Theorem 1.1). The inequality proof combines an affine majorant of f, Fubini's theorem, Schwarz symmetrization, and Brunn-Minkowski equality conditions.

Significance. If the inequality itself is correct, it is a clean extension of the classical Hermite-Hadamard inequality with an explicit and optimal constant for the Santos volume-section question, and the log-concave version is a useful companion estimate. The argument is explicit and does not rely on circular reasoning or fitted parameters. However, the equality characterizations, which are a prominent part of the abstract and of Theorems 1.2 and 1.3, are false as stated for constant functions, and the proof contains an unhandled degenerate case. These are genuine defects in load-bearing statements, though they appear local and repairable.

major comments (3)

[Theorem 1.2 and proof immediately after Eq. (9)] The equality characterization in Theorem 1.2 is false as stated because f≡0 gives equality for every C∈K^n_0: both sides of the inequality are 0, regardless of whether C is a generalized cylinder. The proof excludes this case implicitly when it defines γ=g(0)/δ immediately after Eq. (9), since for f≡0 the affine majorant g has g(0)=0 and δ=0, so γ is undefined. The theorem should either assume f is not identically zero (or f(0)>0) and add a separate trivial case, or correctly describe the full equality class for f≡0.
[Theorem 1.3 and its proof] The same degeneracy invalidates the equality case of Theorem 1.3. If f≡c>0 on C, then fmin=f(0)=c, the right-hand side is c (the quotient tends to 1 as f(0)/fmin tends to 1), and the left-hand side is c, so equality holds for every C∈K^n_0. The proof applies Theorem 1.2 to u-u0≡0, which is exactly the unhandled constant-zero case identified above. The equality statement must exclude constant f or include the full set of equality cases.
[Proof of Theorem 1.2, use of Eq. (9)] The proof of the inequality itself is incomplete for affine majorants with δ=0. For a constant positive f, one may take g=f and then δ=0, so the quantity γ=g(0)/δ used in Eq. (9) is undefined. While the inequality for constant functions is elementary and could be recovered by approximation, the current proof does not provide any argument for this case. The same gap affects the equality proof, which later asserts that equality in (10) forces δ=f(0) without first establishing δ>0.

minor comments (4)

[Abstract] The abstract states f:C->[0,1) and g:[0,1)->[0,1), whereas Theorem 1.2 and the introduction use f:C->[0,∞) and φ:[0,∞)->[0,∞); these codomains should be corrected.
[Equation (10)] In the displayed equation before (10), the integrand is written as (g(0)+tδ/t0)|M_t^{**}|dt, but it should be φ(g(0)+tδ/t0)|M_t^{**}|dt; an application of φ is missing.
[Proof of Theorem 1.2, paragraph after Eq. (9)] The text says M_t'=(te1+L)∩C and (-te1+L)∩C=-2te1+M_t', but M_t' was defined as the sections of C'=σ_{e1}(C), so the intersections should be with C' rather than C.
[Remark 2.1] The example in Remark 2.1 takes f(x)=⟨x,e1⟩-1, which is non-positive on C=[-e1,e1]×C0 and hence not a function with values in [0,∞); it also does not vanish on the basis -e1+C0. The intended example is presumably a positive affine function such as f(x)=c(1+⟨x,e1⟩) with c>0, vanishing on the opposite basis.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main inequalities are derived from standard external theorems (Jensen, Brunn–Minkowski, Schwarz symmetrization) with no fitted parameters or self-citation chain carrying the argument.

full rationale

The paper's central result, Theorem 1.2, is proved directly from classical ingredients: Jensen's inequality and the Hermite–Hadamard inequality, the Brunn–Minkowski inequality (6), and the standard properties of Schwarz symmetrization. The proof constructs an affine majorant g of f, compares integrals over C with integrals over a symmetric cylinder R, and invokes elementary convexity slope comparisons (9) and equality cases of Brunn–Minkowski. No parameter is fitted to the target inequality, and the equality characterization is derived from the equality conditions of the cited external theorems rather than assumed. The applications in Theorem 1.1 and Theorem 1.3 are straightforward substitutions of Corollary 2.2 and Theorem 1.2 respectively, so they inherit the derivation rather than being loaded onto the conclusions. The paper cites two previous works of the author ([AAGJV], [ABG]) only as background references for related inequalities and not as the source of the main theorem's assumptions or conclusion; hence the self-citations are not load-bearing. The noted equality-case gap for constant functions when the supporting affine function has δ = 0 is a genuine local correctness concern about division by δ and the strict-convexity argument, but it is not a circularity: the theorem's inequality still does not presuppose its own conclusion. The derivation chain is self-contained against standard, externally verified results, so the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities appear. The proof relies on standard convex geometry theorems. The equality characterization is not fully supported for constant functions, but this is a gap in the statement, not an extra postulated input.

assumptions (5)

standard math Brunn-Minkowski inequality and its equality case
Used in the equality-case proof of Theorem 1.2 and in the proof of Theorem 1.1 to translate constant section volumes into translational invariance.
standard math Schwarz symmetrization preserves convexity and volume
Used at the start of the proof of Theorem 1.2 to replace C by the body of revolution C' without changing volumes.
standard math Existence of a supporting affine function for a concave function at an interior point
Invoked at the beginning of the proof of Theorem 1.2 to construct g with g(0)=f(0) and g>=f.
standard math Jensen's inequality and the classical Hermite-Hadamard inequality
They provide the baseline and are used in the alpha=1 case of Corollary 2.2 and in Theorem 3.2.
standard math Fubini's theorem
Used throughout to express integrals over C as iterated integrals over sections and projections.

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Pith. "Pith review of On a generalization of the Hermite-Hadamard inequality and applications in convex geometry." pith.science (2026). https://pith.science/paper/TTKOC3SD

@misc{pith2026190806426,
  author       = {Pith},
  title        = {Pith review of: On a generalization of the Hermite-Hadamard inequality and applications in convex geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TTKOC3SD}},
  note         = {Machine review of arXiv:1908.06426}
}

abstract

In this paper we show the following result: if C is an n-dimensional 0-symmetric convex compact set, $f:C\rightarrow[0,1)$ is concave, and $g:[0,1)\rightarrow[0,1)$ is not identically zero, convex, with g(0)=0, then \[ \frac{1}{|C|}\int_C g(f(x))dx \leq \frac12 \int_{-1}^1g(f(0)(1+t))dt, \] where |C| denotes the volume of C. If g? is strictly convex, equality holds if and only if f is affine, C is a generalized symmetric cylinder and f becomes 0 at one of the basis of C. We exploit this inequality to answer a question of Francisco Santos on estimating the volume of a convex set by means of the volume of a central section of it. Second, we also derive a corresponding estimate for log-concave functions.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The complete classification of empty lattice $4$-simplices
math.CO 2019-08 conditional novelty 8.0 of 10

Every empty lattice 4-simplex belongs to one of 49 infinite families or to one of 2461 sporadic simplices with volume at most 419, completing a 30-year classification program.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Alonso-Guti\'errez, S

D. Alonso-Guti\'errez, S. Artstein-Avidan, B. Gonz\'alez Merino, C. H. Jim\'enez, R. Villa, Rogers-Shephard and local Loomis-Whitney type inequalities, Math. Annalen, 374 (2019), no. 3-4, 1719--1771

work page 2019

[2] [2]

Alonso-Guti\'errez, J

D. Alonso-Guti\'errez, J. Bernu\'es, B. Gonz\'alez Merino, An extension of Berwald's inequality and its relation to Zhang's inequality, J. Math. Anal. Appl., 486 (2020), no. 1

work page 2020

[3] [3]

Berwald, Verallgemeinerung eines Mittelwertsatzes von J

L. Berwald, Verallgemeinerung eines Mittelwertsatzes von J. Favard f\"ur positive konkave Funktionen, (German) Acta Math., 79 (1947), 17-–37

work page 1947

[4] [4]

Borell, Complements of Lyapunov's inequality, Math

C. Borell, Complements of Lyapunov's inequality, Math. Ann., 205 (1973), 323–-331

work page 1973

[5] [5]

de la Cal, J

J. de la Cal, J. C\'arcamo, Multidimensional Hermite-Hadamard inequalities and the convex order, J. Math. Anal. Appl., 324 (2006), 248-–261

work page 2006

[6] [6]

S. S. Dragomir, On Hadamard's inequality on a disk, J. Inequal. Pure Appl. Math., 1 (2000), no. 1, 1--11

work page 2000

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S. S. Dragomir, On Hadamard's inequality for the convex mappings defined on a ball in the space and applications,

work page

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S. S. Dragomir, C.E.M. Pearce, Selected Topics on Hermite-Hadamard Inequality and Applications, Victoria University, Melbourne, 2000

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