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Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions

T0 review · 0 major / 2 minor · reviewed 2026-05-24 · grok-4.3

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Pith's one-line read For positive subharmonic functions on convex domains in R^n the volume average is at most 2n^{3/2} times the surface average.

desk verdict The paper sharpens the constant to 2n^{3/2} for positive subharmonic functions on convex domains and adds a clean geometric inequality for nested sets. read the letter →

arxiv 1907.06122 v1 pith:KYV6QB6G submitted 2019-07-13 math.CA math.FAmath.MG

Thomas Beck , Barbara Brandolini , Krzysztof Burdzy , Antoine Henrot , Jeffrey J. Langford , Simon Larson , Robert G. Smits , Stefan Steinerberger This is my paper · ORCID
classification math.CAmath.FAmath.MG
keywords subharmonicfunctionsconvexdomainsHermite-Hadamardinequalityhigherdimensionssurface-to-volumeratiosboundaryintegrals
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

The paper proves an inequality that bounds the average value of a positive subharmonic function inside a convex domain by a multiple of its average on the boundary. The multiple c_n satisfies c_n less than or equal to 2n to the power 3/2 and is at least n minus 1. This extends earlier results that required the stronger assumption that the function itself is convex and used larger constants. The same argument produces a geometric bound: when one convex domain sits inside another the product of their surface-to-volume ratios is at most n.

What carries the argument

The inequality comparing the normalized volume integral of a positive subharmonic function to its normalized surface integral, controlled by a dimension-dependent constant c_n.

What would settle it

A single convex domain together with one positive subharmonic function on it whose volume-to-surface average ratio exceeds 2n^{3/2}.

Watch

Extended reading notes

Core claim

Let Omega subset R^n be convex and let f be positive and subharmonic on Omega. Then the volume average of f is bounded above by c_n times the surface average of f, where c_n is at most 2n^{3/2}. The optimal constant is at least n-1. As a consequence, any two nested convex domains Omega2 subset Omega1 satisfy |partial Omega1|/|Omega1| times |Omega2|/|partial Omega2| less than or equal to n.

Load-bearing premise

The domain must be convex and the function must be positive with nonnegative Laplacian.

Editorial extensions

If this is right

  • The stated bound on c_n holds for all positive subharmonic functions, not merely convex ones.
  • The optimal constant c_n is at least n-1 in every dimension.
  • Nested convex domains obey the product bound on their surface-to-volume ratios.
  • The geometric inequality is sharp in the sense that equality is attained in limiting cases.

Reading between the lines

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

  • The bound may become sharper when the function is harmonic rather than merely subharmonic.
  • The nested-domain inequality could be used to compare isoperimetric ratios across a chain of convex sets.
  • Explicit computation on the Euclidean ball would give a concrete numerical check on how close 2n^{3/2} is to the true optimum.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 2 minor

Summary. The paper claims that for convex Ω ⊂ R^n and positive subharmonic f (Δf ≥ 0), the volume average satisfies (1/|Ω|) ∫_Ω f dx ≤ (c_n / |∂Ω|) ∫_∂Ω f dσ with explicit uniform bound c_n ≤ 2n^{3/2}; the optimal constant satisfies c_n ≥ n-1. It also derives the geometric inequality |∂Ω1|/|Ω1| ⋅ |Ω2|/|∂Ω2| ≤ n whenever Ω2 ⊂ Ω1 are convex. The claims extend Hermite-Hadamard inequalities from convex to subharmonic functions with improved constants.

Significance. If the stated bounds hold, the work supplies explicit, dimension-dependent constants that improve on prior results limited to convex functions, together with a sharp geometric consequence obtained as a byproduct. The approach via the maximum principle for subharmonic functions is consistent with standard tools in the field and yields falsifiable predictions (the lower bound n-1 and the geometric inequality).

minor comments (2)
  1. [Abstract] The abstract states that the inequality 'was previously only known for convex functions with a much larger constant' but supplies neither the prior constant nor a citation; adding this reference would clarify the improvement.
  2. Notation for surface measure (dσ) and volume measure (dx) is standard but should be defined explicitly on first use for readers outside convex geometry.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, accurate summary of our results on improved Hermite-Hadamard bounds for subharmonic functions and the geometric inequality, and the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained

full rationale

The paper presents an inequality for positive subharmonic functions on convex domains together with explicit bounds on the constant c_n and a geometric consequence for nested convex sets. These are derived from the maximum principle for subharmonic functions and scaling properties such as John's theorem; the provided abstract and reader summary contain no fitted parameters renamed as predictions, no self-definitional steps, and no load-bearing self-citations. The central claims reduce to standard analytic inequalities rather than to any of the enumerated circular patterns, so the derivation chain is independent of its own outputs.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The paper relies on standard properties of the Laplacian, convexity, and integration in Euclidean space without introducing new free parameters or postulated entities.

assumptions (2)
  • standard math Standard properties of subharmonic functions (Delta f >= 0) and convex domains in R^n
    Invoked to define the setting and apply mean-value properties.
  • domain assumption Positivity of f
    Required for the inequality as stated in the abstract.

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Cite this review

Pith. "Pith review of Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions." pith.science (2026). https://pith.science/paper/KYV6QB6G

@misc{pith2026190706122,
  author       = {Pith},
  title        = {Pith review of: Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KYV6QB6G}},
  note         = {Machine review of arXiv:1907.06122}
}
abstract

Let $\Omega \subset \mathbb{R}^n$ be a convex domain and let $f:\Omega \rightarrow \mathbb{R}$ be a positive, subharmonic function (i.e. $\Delta f \geq 0$). Then $$ \frac{1}{|\Omega|} \int_{\Omega}{f dx} \leq \frac{c_n}{ |\partial \Omega| } \int_{\partial \Omega}{ f d\sigma},$$ where $c_n \leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_n \geq n-1$. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other $ \Omega_2 \subset \Omega_1 \subset \mathbb{R}^n$: $$ \frac{|\partial \Omega_1|}{|\Omega_1|} \frac{| \Omega_2|}{|\partial \Omega_2|} \leq n.$$

Figures

Figures reproduced from arXiv: 1907.06122 by the authors.

Figure 1
Figure 1. Application of the one-dimensional inequality on a one￾dimensional fiber. This step is lossy if the boundary is curved. Steinhagen [22] showed that width can be bounded in terms of the inradius (7) w(Ω) ≤ ( 2 √ n · inrad(Ω) if n is odd, 2 √n+1 n+2 · inrad(Ω) if n is even. The last inequality follows from [12]: if Ω ⊂ R n is a convex body and Ωt = {x ∈ Ω : d(x, ∂Ω) > t}, where d(x, ∂Ω) denotes the distance to the bou… view at source ↗
Figure 2
Figure 2. The torsion function in Ω is bounded from above by the torsion function of the strip. This shows kukL∞ ≤ w(Ω)2 8 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. The construction of C1 and C2. Since Ω2 ⊂ Ω1, we have that ΩN ∩ {(x, y) ∈ R n : y ≥ 0} = C2 ∩ {(x, y) ∈ R n : y ≥ 0} . We now define a convex function on R n via f(x, y) = ( y if y ≥ 0, 0 otherwise. We obtain Z ΩN f dxdy = Z ΩN ∩{y>0} f dxdy = Z C2∩{y>0} f dxdy = (1 + o(1))N2 2 |Ω2| Z ∂ΩN f dσ = Z ∂ΩN ∩{y>0} f dσ = Z ∂C2∩{y>0} f dσ = (1 + o(1))N2 2 |∂Ω2|. This shows that 1 |ΩN | Z ΩN f dx = (1 + o(1)) 2N |Ω2| |Ω1| a… view at source ↗

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