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The paper establishes sharp one-dimensional entropy analogues of Grünbaum's inequality: left-truncating a centered log-concave variable costs at most a universal constant, and the equality cases are exactly the one-sided exponentials and pi

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For centered one-dimensional log-concave distributions, conditioning on X≤EX can lower differential or min-entropy by at most e/(e-1)H2(1/e) (or log(1+√2)), with sharp equality for mirrored-exponential and piecewise constant-exponential densities.

T0 review reviewed 2026-07-31 challenge →

load-bearing objection The upper-bound half is solid; the reverse differential-entropy inequality rests on an unproved perturbation step that I could not verify. the 2 major comments →

arxiv 2607.23269 v1 pith:JCJ5COPF submitted 2026-07-25 math.PR cs.ITmath.IT

Entropic analogues of Gr\"unbaum's inequality

Matthieu Fradelizi , Lampros Gavalakis , Martin Rapaport This is my paper
classification math.PR cs.ITmath.IT MSC 60E1552A4094A17
keywords Grünbaum inequalitydifferential entropyRényi entropymin-entropylog-concave distributionsentropy inequalitiestruncationsharp constants
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 classical Grünbaum inequality says that a halfspace through the centroid captures at least 1/e of the volume of a convex body. This paper proves the entropy version: for a one-dimensional log-concave distribution, conditioning on a half-line never increases Rényi entropy of any order, and if the cut is at the mean, the entropy loss is bounded by a sharp universal constant — e/(e−1)H2(1/e) for differential entropy and log(1+√2) for min-entropy. The equality cases are the centered exponential for differential entropy and densities that are constant on one side and exponential on the other for min-entropy. The authors also show that in higher dimensions no direct analogue holds without dimension-dependent constants.

Core claim

On the paper's own terms, the central discovery is that Grünbaum's classical 1/e volume-truncation bound has sharp entropic counterparts in dimension one. Theorem 1 states that for every log-concave density f and every threshold m, the Rényi entropy of the left-truncated density f−m is at most the Rényi entropy of f for every order α∈[0,∞], with equality exactly when the support lies to the left of m or f is a one-sided exponential. Theorem 2 adds reverse bounds when m is the mean: hα(f−) ≥ hα(f) − Cα for α=1 and α=∞, with C1=e/(e−1)H2(1/e) and C∞=log(1+√2), and fully characterizes the equality cases. The proof passes through a constrained entropy-minimization result for log-concave densitie

What carries the argument

The load-bearing tools are the entropy decomposition h(X)=p+h(X+)+p−h(X−)+H2(p+) for a truncation, the comparison inequality log ∥g∥∞ ≤ 1 + ∫g log g for log-concave densities (with a new equality characterization), two concentration lemmas bounding first moments of log-concave functions on the half-line, and a reduction of the reverse inequality to a constrained entropy-minimization problem over log-concave densities on R+ with fixed mass, mean, and endpoint condition. That problem is attacked with the degrees-of-freedom method — showing minimizers are log-affine on at most three intervals — followed by Fritz–John stationarity conditions, which force any minimizer to be log-affine on a singl

Load-bearing premise

The lower bound relies on the assumption that the degrees-of-freedom perturbation construction used to reduce the constrained entropy-minimization problem to densities that are log-affine on at most three intervals is valid for the class with endpoint condition f(0)>0 and √f(0) ≥ (log f)'₊(0); if that construction admits too few perturbation directions, the minimizers could be richer and the sharp constants could fail.

What would settle it

Take a centered triangular (or other piecewise-linear log-concave) density and compute h(f)−h(f−) numerically; if it exceeds C1 = e/(e−1)H2(1/e) in nats, the reverse inequality is false. For Proposition 17, run the constrained minimization over compactly supported log-concave densities with mean 1 and the stated endpoint condition for large support length; if the numerical minimizer is not the truncated exponential, the reduction step is wrong.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For a one-dimensional log-concave X and every Rényi order α, the map t↦hα(X|X≤t) is non-decreasing; equivalently, truncating away a tail never increases uncertainty.
  • Combining the min-entropy case with the functional Grünbaum inequality recovers the known pointwise bound ∥f∥∞ ≤ e f(0) for centered log-concave densities on R.
  • The equality characterizations single out the centered exponential family and the piecewise constant-exponential family as the only distributions for which the reverse inequalities are tight.
  • Any direct higher-dimensional extension of the reverse bound must have dimension-dependent constants: for products of centered exponentials cut by a hyperplane through the mean, the entropy gain on the other side grows like √(2n/π).
  • The sharp one-dimensional results provide a benchmark: a Grünbaum-type volume bound recovered from the entropic inequality is suboptimal, so the entropy information is not equivalent to the classical volume statement.

Where Pith is reading between the lines

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

  • The paper leaves open the interpolation of sharp constants for Rényi order α∈(1,∞); one would expect a family of extremizers deforming between the one-sided exponential and the piecewise constant–exponential density, testable by solving the same constrained minimization for general α.
  • The √n entropy gain for product exponentials hints at a central-limit mechanism: the extremal conditioning direction may be the diagonal hyperplane, as conjectured in the paper, linking the question to thin-shell behavior of log-concave measures.
  • The reliability interpretation (decreasing uncertainty of residual life) suggests the one-dimensional result could supply new characterizations and goodness-of-fit tests for exponentiality, since equality in the reverse bound selects exactly the exponential family.
  • The equality characterization of log ∥g∥∞ ≤ 1 + ∫g log g (one-sided and two-sided exponentials) may serve as a rigidity lemma for quantitative versions of the entropy bound, where approximate equality forces approximate exponential shape.
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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

2 major / 4 minor

Summary. The paper proves one-dimensional entropic analogues of Grünbaum's inequality for log-concave distributions. The upper-bound direction (Theorem 1 / Theorem 8) states that for any log-concave density on R, left truncation does not increase Rényi entropy of any order, with equality only for trivial truncation or one-sided exponential laws. The reverse direction (Theorem 2) assumes zero mean and gives sharp lower bounds for the entropy of the left-conditional distribution: the differential entropy can drop by at most (e/(e-1)) H2(1/e), and the min-entropy by at most log(1+sqrt(2)), with complete equality characterizations. The proof of the upper bound is elementary and combines an entropy bound from [22] with a monotonicity argument for the Rényi parameter. The differential-entropy lower bound is proved through a constrained entropy minimization over log-concave densities on R+ (Proposition 17), which is reduced by a degrees-of-freedom argument to log-affine minimizers and then solved by Fritz-John stationarity. The paper also gives high-dimensional counterexamples and discusses possible extensions.

Significance. If correct, these are sharp entropic counterparts to a classical geometric inequality, and the equality characterizations are genuinely informative: they identify the centered exponential (for differential entropy) and a piecewise constant/exponential law (for min-entropy) as extremal. The upper-bound proof is clean and essentially self-contained, and the min-entropy half of Theorem 2 is an elegant application of Borell's inequality and the Milman-Pajor lemma with a complete equality analysis. The paper also provides a useful equality characterization in the comparison inequality log||g||∞ ≤ 1 + ∫ g log g, and its counterexamples in higher dimensions are concrete and rule out naive generalizations. The main weakness is that the differential-entropy lower bound depends on a degrees-of-freedom reduction that is only sketched; this is a load-bearing step and needs a complete proof.

major comments (2)
  1. [Section 3, Proposition 19] The proof of Proposition 19 is the load-bearing step for the differential-entropy lower bound. It asserts that if a minimizer f in F_R is not three-piece log-affine, then [18, Proof of Proposition 2] supplies a nonzero bounded W vanishing near 0 with ∫ Wf = ∫ xWf = 0 such that f(1±tW) are log-concave and lie in F_R for small t. This transfer is not automatic: the cited result concerns convex bodies, while here W must lie in the tangent cone of log-concave densities on [0,R], preserve the endpoint inequality sqrt(f(0)) ≥ u'_+(0), and satisfy the two moment identities after perturbation. The manuscript does not state or prove the dimension of the admissible perturbation space, nor does it treat separately the piecewise-affine and curved cases. Since Proposition 17 and hence Theorem 16 pass through this reduction, the proof is incomplete at this point. Please provide a complete, self-contai
  2. [Section 3, Propositions 17 and 19] The compactness and approximation arguments are also compressed. In Proposition 19, the claims that the normalization ∫ f_n = 1 prevents f_n(0) from tending to 0, that the mean constraint prevents f_n(0) from tending to infinity, and that weak convergence together with [7, Fact 2.5] yields f(0) = lim f_n(0) need to be justified in detail. Similarly, the compact-support approximation at the start of Proposition 17 must show that the endpoint condition and the value of κ are preserved in the limit. These are standard in spirit but not trivial; they should be written out if Proposition 17 is to support Theorem 16.
minor comments (4)
  1. [Section 2.3, Example 12] The density f in Example 12 is unnormalized. The min-entropy inequality is scale-invariant, but this should be stated explicitly to avoid confusion when comparing maxima and integrals.
  2. [Section 3, Theorem 15] After equation (25), the chain '≤ 2p+/(p−)^2 = (1+p−/p+) 2(p+)^2/(p−)^2 ≤ 1+√2' is correct but easy to misread. Spell out the substitution p+ = r p−, r < 1/√2, and the algebra leading to r(1+r) < (1+√2)/2.
  3. [Section 3, Proposition 19] The notation f± := f(1±tW) conflicts with the conditional densities f± used in Theorems 15 and 16. Rename the perturbed densities, for instance f_t^+ and f_t^-, to avoid ambiguity.
  4. [Section 2.2, Proposition 11] In the equality analysis, the sentence 'Comparing these identities for different values of x, we conclude that λ_x does not depend on x' deserves a more explicit derivation. Also define the support endpoint d = sup supp(f) formally, allowing d = +∞.

Circularity Check

0 steps flagged

No circularity: the central inequalities are derived from external prior lemmas and direct optimization; no target result is assumed as an input.

full rationale

The main claims are not circular. Theorem 1's upper bound derives from the known entropy bound log||g||∞≤1+∫g logg of [22], with equality cases handled in Proposition 10; Theorem 2's upper bound is elementary, and its α=1 lower bound passes through the constrained minimization Proposition 17. The only step that is not fully self-contained is the structural reduction in Proposition 19: 'By the one-dimensional degree-of-freedom construction of [18, Proof of Proposition 2], there are enough independent admissible perturbation directions...'. This is a citation to an external perturbation technique rather than to the target inequality or to an equality case of the theorem. The paper then carries out the Fritz–John stationarity argument (Lemma 20) and verifies the desired inequality directly for the resulting one-piece log-affine densities. Likewise, the equality case in Theorem 16 is imported from [19, Theorem 1.4], an external characterization of equality in functional Grünbaum, not from the theorem being proved. The self-citations are to prior results with independent content, not to this paper's conclusions. A legitimate concern is that Proposition 19's transfer of the DoF construction to the constrained class F_R is asserted rather than proved; if the perturbation space were smaller than claimed, the reduction would fail. That is a completeness/rigor issue about a cited lemma, not a circular derivation, since no assumption equivalent to (10) or (8) is used to produce those results. No parameter is fitted to data and no announced constant is assumed.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

No free parameters are fitted. The constants C1 and C∞ are derived, not chosen. The only non-transparent input is the external DoF theorem; the endpoint condition √f(0)≥κ(∫tf)^{1/2} is a derived condition for centered densities, not an ad hoc postulate.

axioms (6)
  • domain assumption Log-concavity of the underlying densities
    Central assumption throughout; all theorems are stated for log-concave f.
  • standard math Entropy bound log||g||∞ ≤ 1 + ∫ g log g for log-concave g (Fradelizi–Meyer [22])
    Used in Lemma 5; equality cases are re-proved in Prop. 10.
  • standard math Degrees-of-freedom method of Fradelizi–Guédon [18]
    Used in Prop. 19 to assert minimizers are log-affine on at most three intervals; not proved in the paper.
  • standard math Fritz–John/Karush–Kuhn–Tucker necessary conditions (Bertsekas [2])
    Used in Prop. 17 to eliminate kinks in candidate minimizers.
  • standard math Borell's inequality and the Milman–Pajor lemma (Lemma 14, Lemma 13)
    Used in Theorem 15 for min-entropy bounds.
  • standard math Functional Grünbaum inequality P(X≥EX)≥1/e ([23,27]) with equality case ([19])
    Used for the no-truncation lower bound and the p-=1/e equality case in Theorem 16.

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Cite this review

Pith. "Pith review of Entropic analogues of Gr\"unbaum's inequality." pith.science (2026). https://pith.science/paper/JCJ5COPF

@misc{pith2026260723269,
  author       = {Pith},
  title        = {Pith review of: Entropic analogues of Gr\"unbaum's inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JCJ5COPF}},
  note         = {Machine review of arXiv:2607.23269}
}
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abstract

The classical Gr\"unbaum inequality asserts that the proportion of the volume of a convex body cut off by a halfspace containing its barycenter is at least $1/e$. From its functional counterpart, for any log-concave random variable $X$, one has $\mathbb{P}(X\ge \mathbb{E}X)\ge 1/e$, with equality if and only if $X$ is exponential. Motivated by Gr\"unbaum's inequality for convex bodies and its functional generalizations, we prove analogous inequalities for entropy, with characterizations of the equality cases. We show that if $X$ is a log-concave random variable on $\mathbb{R}$, then $$ h(X)-\frac{e}{e-1}H_2(1/e) \leq h(X|X \leq \mathbb{E}X) \leq h(X), $$ where $h$ is the differential entropy, $H_2(\cdot)$ is the binary entropy function and $X|X\leq \mathbb{E}X$ stands for the distribution of $X$ conditional on $X\leq \mathbb{E}X$. We generalize the upper bound for all R\'enyi entropies and the lower bound for min-entropy. Our inequalities are sharp and we characterize all equality cases. We discuss potential generalizations in high dimensions and give counterexamples in some directions. As an intermediate step for the proof of the lower bound, we establish a new inequality that we prove using a technique known as degrees of freedom, combined with a standard KKT-type optimization lemma. Along the way, we characterize the equality case in a known comparison inequality between differential and min-entropy, which may be of independent interest.

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This paper was first reviewed by deepseek-v4-flash on July 31, 2026.