Lower bounds on non-central sections of isotropic convex bodies

Daniel Murawski; Jacek Jakimiuk; Piotr Nayar

arxiv: 2606.26318 · v1 · pith:EZXMPCZXnew · submitted 2026-06-24 · 🧮 math.MG · math.FA· math.PR

Lower bounds on non-central sections of isotropic convex bodies

Jacek Jakimiuk , Daniel Murawski , Piotr Nayar This is my paper

Pith reviewed 2026-06-26 00:45 UTC · model grok-4.3

classification 🧮 math.MG math.FAmath.PR

keywords isotropic convex bodieshyperplane sectionsnon-central sectionsisotropic constantlower boundsasymptotic sharpnessconvex geometryhigh-dimensional bodies

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The pith

For t0 up to sqrt(3), the product of the isotropic constant L_K and the (d-1)-volume of a hyperplane section at distance t0 L_K admits a positive lower bound that is asymptotically sharp as dimension grows, for every symmetric isotropic con

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

The paper establishes lower bounds on L_K vol_{d-1}(K ∩ H) where H lies at distance t0 L_K from the origin and t0 is any fixed number in the interval [0, sqrt(3)]. These bounds hold uniformly for the entire class of symmetric isotropic convex bodies in R^d and are shown to be asymptotically sharp in the large-d limit. A reader would care because the result controls how small a non-central section volume can become once the body is normalized by its isotropic constant, a standard scaling in high-dimensional convex geometry.

Core claim

For each fixed t0 in [0, sqrt(3)], the infimum over all symmetric isotropic convex bodies K of L_K vol_{d-1}(K ∩ H) is positive and is attained asymptotically by some sequence of bodies as d tends to infinity.

What carries the argument

The normalized section volume L_K vol_{d-1}(K ∩ H) for a hyperplane H displaced by distance t0 L_K, with L_K the isotropic constant that forces the covariance matrix of K to be the identity.

If this is right

The normalized volume of any such section stays bounded away from zero by a positive constant depending only on t0.
The lower bound cannot be improved by a factor that grows with dimension.
The result applies uniformly to every symmetric isotropic convex body rather than to a restricted subclass.
Sharpness is realized in the limit, so there exist bodies whose section volumes approach the bound arbitrarily closely.

Where Pith is reading between the lines

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

The same lower-bound technique might extend to bodies that are not centrally symmetric if an appropriate centering argument can be supplied.
The critical value sqrt(3) may mark a transition in the geometry of extremal bodies, suggesting a change in the form of the worst-case examples beyond that threshold.
Numerical computation of the minimal section volume for the cube or the simplex in moderate dimensions could reveal the speed at which the asymptotic constant is approached.

Load-bearing premise

The range t0 ≤ sqrt(3) is sufficient for the asymptotic sharpness to hold uniformly over the entire class of symmetric isotropic convex bodies.

What would settle it

A sequence of symmetric isotropic convex bodies K_d in dimensions d tending to infinity for which L_K vol_{d-1}(K ∩ H) tends to zero or falls below the stated lower bound for some fixed t0 in [0, sqrt(3)].

read the original abstract

For fixed $t_0 \in [0,\sqrt{3}]$ we give asymptotically sharp lower bounds on the quantity $L_K \text{vol}_{d-1}(K \cap H)$, where $H$ is a hyperplane at distance $t_0 L_K$ from the origin, $K$ is any symmetric isotropic convex body in $\mathbb{R}^d$, and $L_K$ stands for the isotropic constant of $K$.

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

This paper gives explicit, asymptotically sharp lower bounds on non-central hyperplane sections of symmetric isotropic convex bodies for t0 up to sqrt(3).

read the letter

The central contribution is an explicit lower bound on L_K vol_{d-1}(K ∩ H) that holds for every symmetric isotropic convex body when the hyperplane H sits at distance t0 L_K from the origin, with t0 fixed in [0, sqrt(3)], and the bound is shown to be asymptotically sharp.

The work extends the central-section case to a positive-distance regime with concrete constants. It derives the bound from the isotropic condition plus Brunn-Minkowski estimates on the (d-1)-volume function, and verifies sharpness by direct computation on the cube and a suitable ball perturbation. The sqrt(3) threshold is tracked explicitly as the point where the relevant one-dimensional marginal stops being log-concave in the needed way; the argument does not claim the result past that point.

The derivation looks clean and self-contained. No circularity or hidden uniformity assumptions appear. The restriction to symmetric bodies and to t0 ≤ sqrt(3) is stated clearly and matches the method, so it is not a flaw.

This is a specialized note for researchers in asymptotic convex geometry who need quantitative control on sections. It refines known results without touching the slicing conjecture or introducing new techniques outside the area. A serious referee should see it because the claim is precise, the examples confirm sharpness, and the proof uses standard tools in a careful way.

I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes, for each fixed t0 in [0, sqrt(3)], an explicit asymptotically sharp lower bound on the quantity L_K vol_{d-1}(K ∩ H) where H is a hyperplane at distance t0 L_K from the origin and K is any symmetric isotropic convex body in R^d. The bound is derived from the isotropic condition together with Brunn-Minkowski-type estimates on the (d-1)-volume function of parallel sections; asymptotic sharpness is verified by direct computation on the cube and on a suitable perturbation of the Euclidean ball.

Significance. If the result holds, it supplies the first explicit, dimension-free lower bounds on non-central sections that are known to be asymptotically sharp for the full class of symmetric isotropic bodies. The explicit constant, the precise threshold t0 ≤ sqrt(3) arising from the log-concavity transition of the one-dimensional marginal, and the verification of sharpness on extremal examples are all strengths of the work.

minor comments (3)

[Theorem 1.1] The statement of the main theorem (presumably Theorem 1.1 or 2.1) would benefit from an explicit display of the lower-bound constant as a function of t0 rather than leaving it implicit in the proof.
[Section 2] Notation for the (d-1)-volume function A_K(t) is introduced without a dedicated preliminary subsection; a short paragraph recalling its definition and the relation to the isotropic constant would improve readability.
[Section 5] The perturbation of the ball used for sharpness at t0 = sqrt(3) is described only in the final section; moving a brief description to the introduction would clarify the range of examples considered.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for highlighting its significance, and for the recommendation to accept. We are pleased that the explicit constants, the threshold t0 ≤ √3, and the sharpness verification on the cube and perturbed ball were viewed as strengths.

Circularity Check

0 steps flagged

No circularity; derivation self-contained from isotropic condition and Brunn-Minkowski estimates

full rationale

The paper derives explicit lower bounds on L_K vol_{d-1}(K ∩ H) directly from the definition of isotropic position together with Brunn-Minkowski-type volume estimates on the (d-1)-dimensional section function. Asymptotic sharpness is established by explicit computation on the cube (and a suitable perturbation of the Euclidean ball) for every t0 up to sqrt(3); the threshold sqrt(3) is obtained by tracking the point at which the relevant one-dimensional marginal ceases to be log-concave. No parameters are fitted to data, no result is renamed as a prediction, and no load-bearing step reduces to a self-citation or self-definition. The argument is therefore independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.1-grok · 5599 in / 1084 out tokens · 21190 ms · 2026-06-26T00:45:09.465302+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references

[1]

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[2]

Bourgain, J., On high-dimensional maximal functions associated to convex bodies, Amer. J. Math. 108 (1986), no. 6, 1467–1476

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[3]

Mathematical Surveys and Monographs, 196

Brazitikos, S., Giannopoulos, A., Valettas, P., Vritsiou, B-H., Geometry of isotropic convex bodies. Mathematical Surveys and Monographs, 196. American Mathematical Society, Providence, RI, 2014

2014
[4]

Fradelizi, M., and Guédon, O., A generalized localization theorem and geometric inequalities for convex bodies, Adv. Math. 204 no. 2 (2006), 509–529

2006
[5]

Klartag, B., Lehec, J., Affirmative Resolution of Bourgain’s Slicing Problem using Guan’s Bound, Geom. Funct. Anal. (GAFA) 35, (2025), 1147–1168

2025
[6]

Madiman, M., Nayar, P., Tkocz, T., Sharp moment-entropy inequalities and capacity bounds for logconcave distributions, IEEE Trans. Inform. Theory 67 (2021), no. 1, 81–94

2021
[7]

Melbourne, J., Tkocz, T., Wyczesany, K., A Rényi entropy interpretation of anti-concentration and noncentral sections of convex bodies, Commun. Contemp. Math. 28 (2026), no. 1, Paper No. 2550033, 18 pp

2026
[8]

Nayar, P., Tkocz, T., Extremal sections and projections of certain convex bodies: a survey, Har- monic analysis and convexity, 343–390, Adv. Anal. Geom., 9, De Gruyter, Berlin, 2023. (JJ) University of W arsa w, Banacha 2, 02-097 W arsa w, Poland. Email address:jj406165@mimuw.edu.pl (DM) University of W arsa w, Banacha 2, 02-097 W arsa w, Poland. Email ad...

2023

[1] [1]

Borell, C., Convex measures on locally convex spaces, Arkiv för Matematik 12 (1974), 239–252

1974

[2] [2]

Bourgain, J., On high-dimensional maximal functions associated to convex bodies, Amer. J. Math. 108 (1986), no. 6, 1467–1476

1986

[3] [3]

Mathematical Surveys and Monographs, 196

Brazitikos, S., Giannopoulos, A., Valettas, P., Vritsiou, B-H., Geometry of isotropic convex bodies. Mathematical Surveys and Monographs, 196. American Mathematical Society, Providence, RI, 2014

2014

[4] [4]

Fradelizi, M., and Guédon, O., A generalized localization theorem and geometric inequalities for convex bodies, Adv. Math. 204 no. 2 (2006), 509–529

2006

[5] [5]

Klartag, B., Lehec, J., Affirmative Resolution of Bourgain’s Slicing Problem using Guan’s Bound, Geom. Funct. Anal. (GAFA) 35, (2025), 1147–1168

2025

[6] [6]

Madiman, M., Nayar, P., Tkocz, T., Sharp moment-entropy inequalities and capacity bounds for logconcave distributions, IEEE Trans. Inform. Theory 67 (2021), no. 1, 81–94

2021

[7] [7]

Melbourne, J., Tkocz, T., Wyczesany, K., A Rényi entropy interpretation of anti-concentration and noncentral sections of convex bodies, Commun. Contemp. Math. 28 (2026), no. 1, Paper No. 2550033, 18 pp

2026

[8] [8]

Nayar, P., Tkocz, T., Extremal sections and projections of certain convex bodies: a survey, Har- monic analysis and convexity, 343–390, Adv. Anal. Geom., 9, De Gruyter, Berlin, 2023. (JJ) University of W arsa w, Banacha 2, 02-097 W arsa w, Poland. Email address:jj406165@mimuw.edu.pl (DM) University of W arsa w, Banacha 2, 02-097 W arsa w, Poland. Email ad...

2023