Spectral and Isoperimetric Bounds on Flat Tori

Emanuel Milman

Spectral and Isoperimetric Bounds on Flat Tori

T0 review · 0 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A tile's variance floor controls the spectral gap and isoperimetric constant of any flat torus.

desk verdict A short, honest note that gets the main inequalities right: the 1/12 variance bound is sharp and clean, the isoperimetric extension is solid, and only the unverified ChatGPT remark and an overstrong abstract phrase need attention. read the letter →

arxiv 2608.13052 v1 pith:GDQNVDZ7 submitted 2026-08-13 math.SP math.FAmath.NT

Emanuel Milman This is my paper · ORCID

classification math.SPmath.FAmath.NT MSC 58J5035P1552C07

keywords flattorusspectralgapCheegerconstantcovarianceoffundamentaldomainVoronoicellKLSconjectureslicingproblemlatticegeometry

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 shows that the geometry of a flat torus is controlled from below by the covariance of any measurable tile out of which the torus is built. For every lattice fundamental domain $K$ and every nonzero dual vector $\xi$, the variance of $\langle x,\xi\rangle$ over $K$ is at least $1/12$; consequently the spectral gap satisfies $\lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda)=4\pi^2\lambda_1(\Lambda^*)^2\ge \pi^2/(3\|\operatorname{Cov}_K\|_{\mathrm{op}})$, and the isoperimetric (Cheeger) constant obeys $D_{\mathrm{Che}}(\mathbb{T}_\Lambda)\ge 1/R_1(K)$. When the tile is convex and its covariance is proportional to the identity (isotropic), the recent resolution of the slicing problem yields universal positive lower bounds on the spectral gap, the Cheeger constant, and the shortest nonzero dual vector, independent of dimension -- a flat-torus analogue of the KLS conjecture. The paper also shows that the reverse inequality fails in general, even when the tile is the central Voronoi cell (the set of points no farther from a given lattice point than from any other) and that cell is isotropic, and that a structural sectional tiling condition restores equivalence up to constants.

What carries the argument

The load-bearing object is the covariance matrix $\operatorname{Cov}_K$ of a measurable tile $K$ of the lattice, together with its directional variances and operator norm. The spectral bound rests on the fact that every nontrivial character of the flat torus pushes Haar measure forward to the uniform measure on $\mathbb{R}/\mathbb{Z}$, giving each dual direction a variance floor of $1/12$. The isoperimetric bound uses a directional translation estimate for sets of finite perimeter: the measure of the symmetric difference of a set and its translate by $z$ is bounded by the integral of $|\langle z,\nu_A\rangle|$ over the reduced boundary; averaging this over a fundamental domain and applying Fubini's theorem yields the isoperimetric lower bound in terms of $R_1(K)$, the maximal expected absolute projection of a uniform point of $K$ onto a unit vector. The upper-bound half of Theorem 1.5 uses the slice density of a convex body along a unit direction $\nu$; this density is log-concave by Brunn-Minkowski, the sectional tiling hypothesis identifies its value at zero with $|\xi|$ for the primitive dual vector $\xi$ normal to the tiling hyperplane, and the one-dimensional log-concave estimate $f(0)\le 1/(\sqrt{2}\sigma)$ forces the variance $\sigma^2=\langle\operatorname{Cov}_K\nu,\nu\rangle$ to be at most $1/(2|\xi|^2)$. Finally, the dimension-free lower bounds pass through the recently proved bound on isotropic constants of convex bodies.

What would settle it

Compute $\langle\operatorname{Cov}_K\xi,\xi\rangle$ for any measurable fundamental domain $K$ of $\mathbb{Z}^n$ and a coordinate direction $\xi$; if the value drops below $1/12$, Lemma 1.1 fails, while the paper predicts this never happens and that the unit cube attains equality. For Theorem 1.5, search for a convex fundamental domain satisfying the sectional tiling hypothesis for which $\|\operatorname{Cov}_K\|_{\mathrm{op}}\lambda_1(\Lambda^*)^2$ exceeds $1/(2\delta)$; a numerical search over low-dimensional lattice polytopes would settle whether the stated constants are sharp.

Watch

Extended reading notes

Core claim

The paper's central discovery is a sharp variance estimate for dual directions. Let $\Lambda\subset\mathbb{R}^n$ be a full-rank lattice, $K$ a measurable fundamental domain of finite second moment, and $\xi\in\Lambda^*\setminus\{0\}$. The normalized Lebesgue measure on $K$ pushes forward under the character $x\mapsto \langle x,\xi\rangle \bmod 1$ to Haar measure on $\mathbb{R}/\mathbb{Z}$, so the variance $\langle\operatorname{Cov}_K\xi,\xi\rangle$ is at least the variance of the uniform distribution on $[-1/2,1/2]$, namely $1/12$. Since $\lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda)=4\pi^2\lambda_1(\Lambda^*)^2$, this yields $\lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda)\ge \pi^2/(3\|\operatorname{Cov}_K\|_{\mathrm{op}})$. The same tiling structure, through a directional translation estimate for sets of finite perimeter, gives $D_{\mathrm{Che}}(\mathbb{T}_\Lambda)\ge 1/R_1(K)$. Combining these with the resolved slicing bound on isotropic constants gives a universal $c>0$ such that every unimodular lattice with an isotropic convex fundamental domain (in particular an isotropic Voronoi cell) satisfies $D_{\mathrm{Che}}(\mathbb{T}_\Lambda),\lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda),\lambda_1(\Lambda^*)\ge c$. There exist lattices with isotropic Voronoi cells for which no dimension-free upper bound holds; under the sectional tiling hypothesis, however, $\|\operatorname{Cov}_K\|_{\mathrm{op}}\lambda_1(\Lambda^*)^2$ is confined to the interval $[1/12,1/(2\delta)]$.

Load-bearing premise

All dimension-free converse statements rest on the sectional tiling hypothesis: a convex fundamental domain with barycenter at the origin must have a linear hyperplane section that tiles that hyperplane by lattice translations and whose normal-direction variance is a fixed fraction of the maximal variance; without this hypothesis, the paper shows the matching upper bound can fail.

Editorial extensions

If this is right

From the covariance of any measurable tile one obtains a dimension-free lower bound on the spectral gap and Cheeger constant of the corresponding flat torus, so bounding a tile's covariance certifies a spectral gap.
For every unimodular lattice whose Voronoi cell is isotropic, the torus satisfies the KLS-type conclusion: $D_{\mathrm{Che}}(\mathbb{T}_\Lambda)$, $\lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda)$, and $\lambda_1(\Lambda^*)$ are all bounded below by a universal constant independent of dimension.
Isotropy alone does not give a matching upper bound: there are lattices with isotropic Voronoi cells whose torus spectral gap grows like a positive power of the dimension.
Under the sectional tiling hypothesis, $\|\operatorname{Cov}_K\|_{\mathrm{op}}\lambda_1(\Lambda^*)^2$ lies between $1/12$ and $1/(2\delta)$, so the torus spectral gap and the Neumann spectral gap of the convex tile are comparable up to a $\sqrt{\log(1+n)}$ factor.
For convex tiles satisfying the sectional tiling condition, covariance bounds and spectral bounds are interchangeable up to constants.

Reading between the lines

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

Editorial inference: the variance inequality behaves like an uncertainty principle for lattices -- no measurable tile can place less than $1/12$ variance on any dual direction; a natural test is whether an analogous floor holds for tilings by non-convex tiles in other periodic geometries.
Editorial inference: the sectional tiling condition is exactly the missing ingredient that upgrades a one-sided covariance bound to an equivalence; it would be worth investigating how common such tiling sections are among natural families of fundamental domains such as zonotopes and alcoved polytopes.
Editorial inference: the universal constant $c$ in Corollary 1.3 is not made explicit; tracking the constants through the slicing bound and the log-concave density estimates would produce a concrete numerical value, which the paper does not provide.
Editorial inference: the existence of isotropic Voronoi cells with spectral gap growing like $\sqrt{n}$ suggests an extremal question -- maximize $\lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda)$ among unimodular lattices with isotropic Voronoi cell; the Hermite-constant bound places this maximum between order $\sqrt{n}$ and order $n$.

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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

Short version: this is a clean, honest note that gets the main things right. Lemma 1.1 is a genuinely nice observation—the uniform mod-1 argument giving the sharp 1/12 bound is airtight, and the spectral gap consequence is immediate. The isoperimetric half is also in good shape: the BV proof of Hadwiger's bound extends it to all lattices, and the application to the flat-torus KLS analogue via Klartag–Lehec is legitimate, not forced. Theorem 1.5's sectional tiling criterion is new and the proof is checkable; the explicit delta-dependence is a plus.

Soft spots, in proportion. Remark 4.2 is a real problem: it is ChatGPT-attributed, explicitly unverified, and contains an undefined symbol p. It should be proved, removed, or replaced with a citation. The abstract's phrase "equivalence up to numerical constants" overstates Theorem 1.5, since the upper bound is 1/(2δ) and δ can be as small as 1/n for the standard cube, so the ratio grows with dimension. That is a presentation-level overreach, not a flaw in the lower bound, but it should be rephrased. The AI Declaration is unusually transparent, but it leaves reproducibility questions only for Remark 4.2 and the counterexample discussion—the main theorems are proven in the text.

I checked the Barnes–Wall example and the Autissier–Magazinov/Siegel counterexample: both look correct. Corollary 1.4 is fine. No load-bearing flaw anywhere. The reader's strongest claim holds up on reading.

This paper is for lattice theorists, spectral geometers, and people in high-dimensional probability who want sharp, explicitly stated constants. It deserves a serious referee and, after a minor revision that cleans up Remark 4.2 and the abstract, it should be accepted.

Referee Report

0 major / 4 minor

Summary. This note studies the flat torus T_Λ=R^n/Λ and relates spectral and isoperimetric parameters to the covariance structure of a fundamental domain K. Lemma 1.1 shows that for every measurable fundamental domain K with finite second moment and every nonzero dual vector ξ, ⟨Cov_K ξ,ξ⟩ ≥ 1/12; consequently λ_SG(T_Λ)=4π²λ_1(Λ*)² ≥ π²/(3‖Cov_K‖_op). Theorem 1.2 (after Hadwiger) proves I_{T_Λ}(v) ≥ 2v(1−v)/R_1(K), hence D_Che(T_Λ) ≥ 1/R_1(K). These bounds yield Corollary 1.3: for unimodular lattices with isotropic Voronoi cell, λ_1(Λ*), λ_SG(T_Λ), and D_Che(T_Λ) are bounded below by a universal constant, using the Klartag–Lehec resolution of the slicing problem. Corollary 1.4 shows that no dimension-free converse of (0.1) holds for general Voronoi cells, and Example 4.1 gives a family of isotropic Voronoi cells with unbounded spectral gap. Theorem 1.5 proves a converse under a sectional tiling hypothesis, giving ‖Cov_K‖_op λ_1(Λ*)² ≤ 1/(2δ). The paper explicitly marks Remark 4.2 as an unverified ChatGPT-attributed claim.

Significance. The main lower-bound argument is elementary, elegant, and checkable: Lemma 1.1 uses the uniform mod-1 distribution and gives the sharp parameter-free constant 1/12, and Theorem 1.2 is proved by a complete BV translation estimate and averaging argument. The connection between the KLS conjecture and flat tori via the Klartag–Lehec slicing theorem is a clean and appealing observation. The central lower-bound theorems and the conditional upper bound in Theorem 1.5 are sound. The paper is also transparent about provenance and limitations. I found no load-bearing error in the main proofs; the issues are local notational and presentational, plus an unverified remark that should be removed or rigorously supported.

minor comments (4)

[Example 4.1, §4] The notational convention "put n = 2m" is inconsistent with the displayed equality λ_1(BW_m)^2 = 2^{(m−1)/2} = √(n/2): for n = 2m, √(n/2) = √m, which is not equal to 2^{(m−1)/2} for m ≥ 3 (e.g., m=3 gives 2 versus √3). The standard Barnes–Wall lattice BW_m has dimension 2^m, and with n = 2^m the displayed equality is correct. Please correct "n=2m" to "n=2^m" (and similarly the phrase "representation on R^{2m}"), or define the lattice family precisely with a reference.
[Remark 4.2, §4] This remark contains an unverified claim attributed to ChatGPT, uses an undefined symbol p, and does not specify how p depends on n. The assertion that this is "essentially the maximal growth rate" is not proved in the text. Since the remark is not used in any proof and Example 4.1 already supplies the necessary counterexample, I recommend deleting it or replacing it with a rigorously verified statement or a precise citation.
[Abstract and §1.5] The abstract's phrase "equivalence (up to numerical constants)" is stronger than what Theorem 1.5 establishes: the upper bound carries the factor 1/(2δ), and when δ is not a universal constant (e.g., δ=1/n for the standard cube) the multiplicative gap between the upper and lower bounds grows with dimension. Please rephrase as "up to constants depending on δ" or explicitly assume δ is a universal constant.
[§1.3, Eq. (1.6) and Ref. [19]] The text ascribes to [19] the rate c log(1+n)^{-1/2}, but the title of [19] states O(log^{1/4} n). Please reconcile the quoted rate with the cited source.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: Lemma 1.1 and Theorem 1.5 are self-contained derivations with no fitted inputs; the only minor self-citation (Ref. [23]) is background and not load-bearing.

full rationale

The paper's central chain is not circular. Lemma 1.1 derives the directional variance lower bound from the Haar-pushforward of the torus character, which is a first-principles fact, and Eq. (1.2) follows by the elementary inequality Var(Y) <= ||Cov_K||_op |xi|^2. Theorem 1.2 is a direct consequence of the proven BV translation estimate (Lemma 3.1) and the averaging identity over K; no parameter is fitted. Theorem 1.5 assumes the sectional tiling hypothesis explicitly and derives f_Y(0)=|xi| from Lemma 5.1, then applies Hensley's one-dimensional log-concave bound; the delta-dependent upper bound is exactly what the hypothesis costs, so the result is not equivalent to its input by construction. Corollary 1.3 relies on the external Klartag-Lehec resolution of Bourgain's slicing problem, which is independent evidence and not a self-citation. The only self-citation is Ref. [23] for the standard equivalence between the KLS conjecture and a Neumann spectral-gap formulation; that is background and non-load-bearing. Presentation limitations (the 'equivalence' in Theorem 1.5 retains a delta-dependence, and Remark 4.2 contains an unverified ChatGPT-attributed claim) are correctness and clarity issues, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It relies on several external theorems from convex geometry, lattice theory, and spectral geometry, listed above, plus the standard tools of BV theory and Brunn-Minkowski theory used inside proofs.

assumptions (6)

domain assumption Klartag-Lehec resolution of Bourgain's Slicing Problem: every centered convex body K in R^n has isotropic constant L_K <= C for a universal constant C.
Used in Corollary 1.3 to bound ||Cov_{K_Lambda}||_op from above for isotropic Voronoi cells with det Lambda = 1.
standard math Log-concave density estimate (5.8): for a log-concave probability density on R with mean 0 and variance sigma^2, 1/(sqrt(12) sigma) <= f(0) <= 1/(sqrt(2) sigma).
Used in the proof of Theorem 1.5 to relate the density at 0 (equal to |xi| by the tiling condition) to the directional variance sigma^2.
standard math Cheeger inequality and De Ponti-Mondino's Buser-type inequality: (1/4) D_Che(T)^2 <= lambda_SG(T) <= pi D_Che(T)^2 for flat tori.
Connects isoperimetric and spectral parameters; used to derive Corollary 1.3 and to compare Lemma 1.1 and Theorem 1.2.
standard math Concavity and symmetry of the isoperimetric profile on flat tori (Sternberg-Zumbrun, Bayle).
Ensures D_Che(T_Lambda) = 2 I_T_Lambda(1/2), used to state the Cheeger bound and discuss sharpness.
standard math Autissier-Magazinov second-moment bound for lattice Voronoi cells, and Siegel's mean value theorem.
Used in Corollary 1.4 to construct lattices with no dimension-free converse to Lemma 1.1.
domain assumption Barnes-Wall lattice properties: BW_m is even unimodular with lambda_1(BW_m)^2 = 2^{(m-1)/2}, and its automorphism group acts irreducibly on R^{2m}.
Used in Example 4.1 to produce isotropic Voronoi cells with dimension-dependent spectral gap.

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Cite this review

Pith. "Pith review of Spectral and Isoperimetric Bounds on Flat Tori." pith.science (2026). https://pith.science/paper/GDQNVDZ7

@misc{pith2026260813052,
  author       = {Pith},
  title        = {Pith review of: Spectral and Isoperimetric Bounds on Flat Tori},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GDQNVDZ7}},
  note         = {Machine review of arXiv:2608.13052}
}

abstract

We record several elementary relations between spectral and isoperimetric parameters of a flat torus $\mathbb{T}_\Lambda = \mathbb{R}^n/\Lambda$ and the covariance structure of a fundamental domain $K$ for the lattice $\Lambda \subset \mathbb{R}^n$. For every measurable fundamental domain $K$ and nonzero vector $\xi$ in the dual lattice $\Lambda^*$, we observe the sharp directional variance estimate \[ \left\langle \operatorname{Cov}_K \xi,\xi\right\rangle \geq \frac{1}{12}. \] This yields a lower bound on the torus spectral gap $\lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda)$ (equivalently, the length of the shortest nonzero dual vector $\lambda_1(\Lambda^*)$) in terms of the maximal covariance of $K$: \[ \lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda) = 4\pi^2 \lambda_1(\Lambda^*)^2 \geq \frac{\pi^2}{3\left\|\operatorname{Cov}_K\right\|_{\mathrm{op}}}. \] Analogous sharp results are obtained for the isoperimetric profile and the Cheeger constant $D_{\mathrm{Che}}(\mathbb{T}_\Lambda)$ using an old argument of Hadwiger. In particular, when the Voronoi cell $K_\Lambda$ of a lattice with $\det \Lambda = 1$ is isotropic, the recent resolution of the Slicing Problem by Klartag and Lehec implies that \[ D_{\mathrm{Che}}(\mathbb{T}_\Lambda),\quad \lambda_{\mathrm{SG}}(\mathbb{T}_\Lambda),\quad \lambda_1(\Lambda^*) \geq c > 0, \] where $c > 0$ is a universal constant independent of dimension $n$; this may be thought of as a positive resolution of the Kannan--Lov\'asz--Simonovits conjecture for all flat tori. While there are lattices $\Lambda$ and corresponding Voronoi cells $K = K_\Lambda$ for which no dimension-independent converse inequality to the spectral-gap bound above can hold, we show that under a certain sectional tiling hypothesis, this inequality is in fact an equivalence (up to numerical constants).

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Works this paper leans on

28 extracted references · 25 canonical work pages

[1]

An improved constant in Banaszczyk's transference theorem

D. Aggarwal and N. Stephens-Davidowitz,An improved constant in Banaszczyk’s transference theorem, arXiv:1907.09020 (2019)

work page Pith review arXiv 2019
[2]

Artstein-Avidan, A

S. Artstein-Avidan, A. Giannopoulos and V. D. Milman,Asymptotic Geometric Analysis, Part I, Mathematical Surveys and Monographs, vol. 202, Amer. Math. Soc., Providence, RI, 2015

work page 2015
[3]

Autissier, Un lemme matriciel effectif,Math

P. Autissier, Un lemme matriciel effectif,Math. Z.273(2013), 355–361

work page 2013
[4]

Bayle,A differential inequality for the isoperimetric profile, Int

V. Bayle,A differential inequality for the isoperimetric profile, Int. Math. Res. Not.2004 (2004), no. 7, 311–342. 9

work page 2004
[5]

Bizeul,OptimalMM ∗ bounds for convex bodies, arXiv:2607.29458 (2026)

P. Bizeul,OptimalMM ∗ bounds for convex bodies, arXiv:2607.29458 (2026)

arXiv 2026
[6]

Buser,A note on the isoperimetric constant, Ann

P. Buser,A note on the isoperimetric constant, Ann. Sci. ´Ecole Norm. Sup. (4)15(1982), no. 2, 213–230

work page 1982
[7]

Cheeger,A lower bound for the smallest eigenvalue of the Laplacian, in: Problems in Analysis (Papers dedicated to Salomon Bochner), Princeton Univ

J. Cheeger,A lower bound for the smallest eigenvalue of the Laplacian, in: Problems in Analysis (Papers dedicated to Salomon Bochner), Princeton Univ. Press, Princeton, NJ, 1970, 195–199

work page 1970
[8]

Chen and B

Y. Chen and B. Klartag,Digesting the proof of the sharp thin-shell inequality, arXiv:2607.23307 (2026)

arXiv 2026

Show all 28 references

[9]

De Ponti and A

N. De Ponti and A. Mondino,Sharp Cheeger–Buser type inequalities in RCD(K,∞ )spaces, J. Geom. Anal.31(2021), 2416–2438

2021
[10]

Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem,J

B. Fuglede, Commuting self-adjoint partial differential operators and a group theoretic problem,J. Funct. Anal.16(1974), no. 1, 101–121

1974
[11]

Fradelizi,Hyperplane sections of convex bodies in isotropic position, Beitr

M. Fradelizi,Hyperplane sections of convex bodies in isotropic position, Beitr. Algebra Geom. 40(1999), no. 1, 163–183

1999
[12]

P. M. Gruber and C. G. Lekkerkerker,Geometry of Numbers, 2nd ed., North-Holland, Amsterdam, 1987

1987
[13]

Hadwiger,Gitterperiodische Punktmengen und Isoperimetrie, Monatsh

H. Hadwiger,Gitterperiodische Punktmengen und Isoperimetrie, Monatsh. Math.76(1972), 410–418

1972
[14]

Hensley,Slicing convex bodies—bounds for slice area in terms of the body’s covariance, Proc

D. Hensley,Slicing convex bodies—bounds for slice area in terms of the body’s covariance, Proc. Amer. Math. Soc.79(1980), no. 4, 619–625

1980
[15]

Hoshikawa and H

Y. Hoshikawa and H. Urakawa, Affine Weyl groups and the boundary value eigenvalue problems of the Laplacian,Interdiscip. Inform. Sci.16(2010), no. 1, 93–109

2010
[16]

Kannan, L

R. Kannan, L. Lov´ asz and M. Simonovits,Isoperimetric problems for convex bodies and a localization lemma, Discrete Comput. Geom.13(1995), no. 3–4, 541–559

1995
[17]

Klartag,Logarithmic bounds for isoperimetry and slices of convex sets, Ars Inven

B. Klartag,Logarithmic bounds for isoperimetry and slices of convex sets, Ars Inven. Anal. (2023), Paper No. 4, 17 pp

2023
[18]

Klartag and J

B. Klartag and J. Lehec,Affirmative resolution of Bourgain’s slicing problem using Guan’s bound, Geom. Funct. Anal.35(2025), 1147–1168

2025
[19]

Letwin,The KLS constant isO(log 1/4n), arXiv:2607.24164 (2026)

B. Letwin,The KLS constant isO(log 1/4n), arXiv:2607.24164 (2026)

2026 arXiv
[20]

Magazinov,A proof of a conjecture by Haviv, Lyubashevsky and Regev on the second moment of a lattice Voronoi cell, Adv

A. Magazinov,A proof of a conjecture by Haviv, Lyubashevsky and Regev on the second moment of a lattice Voronoi cell, Adv. Geom.20(2020), no. 1, 117–120

2020
[21]

Maggi,Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, Cambridge Studies in Advanced Mathematics, vol

F. Maggi,Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, Cambridge Studies in Advanced Mathematics, vol. 135, Cambridge Univ. Press, Cambridge, 2012

2012
[22]

Melbourne, M

J. Melbourne, M. Roysdon, C. Tang and T. Tkocz,From simplex slicing to sharp reverse H¨ older inequalities, J. Lond. Math. Soc. (2)113(2026), no. 3, Paper No. e70461, 23 pp. 10

2026
[23]

Milman, On the role of convexity in isoperimetry, spectral-gap and concentration,Invent

E. Milman, On the role of convexity in isoperimetry, spectral-gap and concentration,Invent. Math.177(2009), no. 1, 1–43

2009
[24]

Nebe, The second minimum of Barnes–Wall lattices, arXiv:2603.23133 (2026)

G. Nebe, The second minimum of Barnes–Wall lattices, arXiv:2603.23133 (2026)

2026 arXiv
[25]

G. Nebe, E. M. Rains and N. J. A. Sloane, The invariants of the Clifford groups,Des. Codes Cryptogr.24(2001), 99–121

2001
[26]

Ros,The isoperimetric problem, in: Global Theory of Minimal Surfaces, Clay Math

A. Ros,The isoperimetric problem, in: Global Theory of Minimal Surfaces, Clay Math. Proc. 2, Amer. Math. Soc., Providence, RI, 2005, 175–209

2005
[27]

C. L. Siegel,A mean value theorem in geometry of numbers, Ann. of Math. (2)46(1945), no. 2, 340–347

1945
[28]

Sternberg and K

P. Sternberg and K. Zumbrun,On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint, Comm. Anal. Geom.7(1999), no. 1, 199–220. 11

1999

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

An improved constant in Banaszczyk's transference theorem

D. Aggarwal and N. Stephens-Davidowitz,An improved constant in Banaszczyk’s transference theorem, arXiv:1907.09020 (2019)

work page Pith review arXiv 2019

[2] [2]

Artstein-Avidan, A

S. Artstein-Avidan, A. Giannopoulos and V. D. Milman,Asymptotic Geometric Analysis, Part I, Mathematical Surveys and Monographs, vol. 202, Amer. Math. Soc., Providence, RI, 2015

work page 2015

[3] [3]

Autissier, Un lemme matriciel effectif,Math

P. Autissier, Un lemme matriciel effectif,Math. Z.273(2013), 355–361

work page 2013

[4] [4]

Bayle,A differential inequality for the isoperimetric profile, Int

V. Bayle,A differential inequality for the isoperimetric profile, Int. Math. Res. Not.2004 (2004), no. 7, 311–342. 9

work page 2004

[5] [5]

Bizeul,OptimalMM ∗ bounds for convex bodies, arXiv:2607.29458 (2026)

P. Bizeul,OptimalMM ∗ bounds for convex bodies, arXiv:2607.29458 (2026)

arXiv 2026

[6] [6]

Buser,A note on the isoperimetric constant, Ann

P. Buser,A note on the isoperimetric constant, Ann. Sci. ´Ecole Norm. Sup. (4)15(1982), no. 2, 213–230

work page 1982

[7] [7]

Cheeger,A lower bound for the smallest eigenvalue of the Laplacian, in: Problems in Analysis (Papers dedicated to Salomon Bochner), Princeton Univ

J. Cheeger,A lower bound for the smallest eigenvalue of the Laplacian, in: Problems in Analysis (Papers dedicated to Salomon Bochner), Princeton Univ. Press, Princeton, NJ, 1970, 195–199

work page 1970

[8] [8]

Chen and B

Y. Chen and B. Klartag,Digesting the proof of the sharp thin-shell inequality, arXiv:2607.23307 (2026)

arXiv 2026