Spectral and Isoperimetric Bounds on Flat Tori

Abstract: We record several elementary relations between spectral and isoperimetric parameters of a flat torus $\mathbb{T}<em>Λ= \mathbb{R}<sup>n/Λ$ and the covariance structure of a fundamental domain $K$ for the lattice $Λ\subset \mathbb{R}<sup>n$. For every measurable fundamental domain $K$ and nonzero vector $ξ$ in the dual lattice $Λ<sup>*$, we observe the sharp directional variance estimate [ \left\langle \operatorname{Cov}_K ξ,ξ\right\rangle \geq \frac{1}{12}. ] This yields a lower bound on the torus spectral gap $λ</em>{\mathrm{SG}}(\mathbb{T}<em>Λ)$ (equivalently, the length of the shortest nonzero dual vector $λ_1(Λ<sup>*)$) in terms of the maximal covariance of $K$: [ λ{\mathrm{SG}}(\mathbb{T}Λ) = 4π^2 λ_1(Λ^*)^2 \geq \frac{π^2}{3\left|\operatorname{Cov}_K\right|{\mathrm{op}}}. ] Analogous sharp results are obtained for the isoperimetric profile and the Cheeger constant $D_{\mathrm{Che}}(\mathbb{T}<em>Λ)$ using an old argument of Hadwiger. In particular, when the Voronoi cell $K</em>Λ$ of a lattice with $\det Λ= 1$ is isotropic, the recent resolution of the Slicing Problem by Klartag and Lehec implies that [ D_{\mathrm{Che}}(\mathbb{T}Λ),\quad λ{\mathrm{SG}}(\mathbb{T}Λ),\quad λ_1(Λ^*) \geq c > 0, ] where $c &gt; 0$ is a universal constant independent of dimension $n$; this may be thought of as a positive resolution of the Kannan--Lovász--Simonovits conjecture for all flat tori. While there are lattices $Λ$ and corresponding Voronoi cells $K = K</em>Λ$ 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).

• The paper proves that every nonzero dual-lattice direction has variance at least 1/12, yielding the sharp bound λSG(TΛ) ≥ π²/(3 CovKop) for any measurable fundamental domain.
• The paper uses Hadwiger’s translation method and log-concavity to bound the Cheeger constant by 1/R₁(K), and confirms the KLS inequality for flat tori using the resolution of Bourgain’s Slicing Problem.
• The paper shows that no dimension-free converse generally holds—even for isotropic Voronoi cells—while a sectional tiling condition restores two-sided spectral–covariance equivalence up to constants.

Overview

This note by Milman establishes quantitative relations between spectral and isoperimetric parameters of a flat torus $T_\Lambda = \mathbb{R}^n/\Lambda$ and the covariance structure of a fundamental domain $K$ for the lattice $\Lambda$. The central observation is elementary but sharp: every nonzero dual lattice vector has variance at least $1/12$ under the uniform measure on any measurable fundamental domain. This yields a lower bound on the spectral gap $$\lambda_{SG}(T_\Lambda) = 4\pi^2\lambda_1(\Lambda^*)^2$$ in terms of the operator norm of $Cov_K$, and, via an argument going back to Hadwiger, an analogous bound on the isoperimetric profile and Cheeger constant. Combined with the recent resolution of Bourgain's Slicing Problem by Klartag and Lehec [KlartagLehec2025], these bounds give a positive resolution of the Kannan–Lovász–Simonovits (KLS) isoperimetric conjecture restricted to flat tori. The paper also records counterexamples showing that no dimension-free converse holds in general, and identifies a sectional tiling hypothesis under which the main inequality becomes an equivalence up to constants.

A sharp directional variance estimate

The key lemma states that if $K$ is any measurable fundamental domain of $\Lambda$ with finite second moment and $\xi \in \Lambda^*$ is nonzero, then

$\langle Cov_K\,\xi,\xi\rangle \geq \frac{1}{12}.$

The proof is a direct Haar-measure computation: since $K$0 tiles $K$1 by $K$2-translates, $K$3 is Haar-uniform on $K$4 when $K$5 is uniform on $K$6. The character $K$7 is a surjective continuous homomorphism, so it pushes Haar measure forward to Haar measure on $K$8; the centered representative of $K$9 is then uniform on $\Lambda$0, whose second moment is exactly $\Lambda$1. Choosing a shortest nonzero dual vector gives

$\Lambda$2

The constant $\Lambda$3 is sharp, attained by $\Lambda$4, $\Lambda$5, and coordinate directions. Notably, the paper's AI declaration credits ChatGPT with observing this sharp constant, replacing a prior weaker universal constant valid only for convex $\Lambda$6.

An important caveat: the bound holds for any fundamental domain, but different domains yield very different estimates — e.g., for $\Lambda$7, the parallelogram generated by $\Lambda$8 and $\Lambda$9 has maximal covariance of order $1/12$0. The Voronoi cell $1/12$1 is presumably always the right choice, though this preference is not proved in general.

Isoperimetric bounds via Hadwiger's theorem

The isoperimetric analogue uses the parameter $1/12$2, the circumradius of the $1/12$3-centroid body of $1/12$4: the supremum over unit directions $1/12$5 of $1/12$6. Hadwiger's theorem (originally stated for $1/12$7; the proof is reproduced here because the original text is in German) asserts

$1/12$8

The proof avoids Crofton-type formulas and works directly for general sets of finite perimeter. It rests on a directional BV translation estimate: $1/12$9, established by mollification of periodic BV functions. Averaging translations over $$\lambda_{SG}(T_\Lambda) = 4\pi^2\lambda_1(\Lambda^*)^2$$0 yields $$\lambda_{SG}(T_\Lambda) = 4\pi^2\lambda_1(\Lambda^*)^2$$1 — the probability that two independent Haar-uniform points lie on opposite sides of $$\lambda_{SG}(T_\Lambda) = 4\pi^2\lambda_1(\Lambda^*)^2$$2 — and Fubini completes the argument. For $$\lambda_{SG}(T_\Lambda) = 4\pi^2\lambda_1(\Lambda^*)^2$$3, one has $$\lambda_{SG}(T_\Lambda) = 4\pi^2\lambda_1(\Lambda^*)^2$$4, giving a sharp Cheeger constant for the standard flat torus and, by reflection, for the unit cube.

When $$\lambda_{SG}(T_\Lambda) = 4\pi^2\lambda_1(\Lambda^*)^2$$5 has barycenter at the origin, Jensen's inequality gives $$\lambda_{SG}(T_\Lambda) = 4\pi^2\lambda_1(\Lambda^*)^2$$6, so the isoperimetric bound implies the spectral bound up to constants; for convex $$\lambda_{SG}(T_\Lambda) = 4\pi^2\lambda_1(\Lambda^*)^2$$7 the two are equivalent up to numeric constants, using $$\lambda_{SG}(T_\Lambda) = 4\pi^2\lambda_1(\Lambda^*)^2$$8 [MelbourneRoysdonTangTkocz2026].

The KLS conjecture on flat tori

The KLS conjecture [KLS1995] posits that every convex body satisfies $$\lambda_{SG}(T_\Lambda) = 4\pi^2\lambda_1(\Lambda^*)^2$$9 with dimension-independent $Cov_K$0. Via the Cheeger–Buser inequalities (with De Ponti–Mondino's improved constant under non-negative Ricci curvature), this is equivalent to a matching lower bound on the Neumann spectral gap. The conjecture reduces to isotropic bodies, where it is controlled by the isotropic constant $Cov_K$1; Klartag–Lehec's resolution of Bourgain's Slicing Problem gives $Cov_K$2 universally.

The results above confirm the KLS inequality for all flat tori, with respect to the covariance of any fundamental domain. The cleanest formulation: if $Cov_K$3 and the Voronoi cell $Cov_K$4 is isotropic, then $Cov_K$5, $Cov_K$6, and $Cov_K$7 are all bounded below by a universal constant independent of dimension. Any isotropic convex fundamental domain would serve equally well. For context, the full KLS conjecture is now known up to a factor $Cov_K$8 [Klartag2023, ChenKlartag2026, Letwin2026], so the toral result is strictly stronger than what is currently available for general convex bodies.

Counterexamples to a converse

A natural reverse inequality $Cov_K$9 fails badly. Combining the Autissier–Magazinov second-moment estimate for Voronoi cells ($K$0) with Siegel mean-value bounds on the covering radius relative to the dual systole ($K$1), there exist lattices $K$2 with

$K$3

so no dimension-free converse can hold for arbitrary Voronoi cells.

More strikingly, even isotropy of the Voronoi cell does not force a uniformly bounded spectral gap from above. The Barnes–Wall lattices $K$4 have isotropic Voronoi cells — irreducibility of the extraspecial Clifford subgroup of their automorphism group forces $K$5 to be a scalar matrix by Schur's lemma — yet, being unimodular and self-dual with $K$6,

$K$7

which grows without bound. The paper attributes (without verification) an even faster example to ChatGPT: duals of Craig cyclotomic lattices, with $K$8 along a subsequence of dimensions. This is essentially maximal, since the Hermite constant satisfies $K$9, so $\Lambda$0 for all unimodular $\Lambda$1 regardless of isotropy.

Equivalence under sectional tiling

The positive converse requires structure. Suppose $\Lambda$2 is a convex fundamental domain with barycenter at the origin, and there exists a hyperplane $\Lambda$3 such that (i) $\Lambda$4 tiles $\Lambda$5 by translations of $\Lambda$6, and (ii) $\Lambda$7 for some $\Lambda$8. Then

$\Lambda$9

so the spectral-gap/covariance relation becomes a two-sided equivalence up to constants.

The proof combines three ingredients. First, tiling forces $\xi \in \Lambda^*$0 for some primitive $\xi \in \Lambda^*$1, and the covolume identity $\xi \in \Lambda^*$2 follows from a fundamental-prism volume count. Second, the density $\xi \in \Lambda^*$3 of $\xi \in \Lambda^*$4 for $\xi \in \Lambda^*$5 uniform on $\xi \in \Lambda^*$6 is log-concave (Brunn–Minkowski) with mean zero, so Hensley–Fradelizi bounds give $\xi \in \Lambda^*$7. Third, the sectional tiling hypothesis makes $\xi \in \Lambda^*$8 exactly. Together with assumption (ii), this yields the upper bound.

Via the near-resolution of KLS, this gives an almost-equivalence between $\xi \in \Lambda^*$9 and $\langle Cov_K\,\xi,\xi\rangle \geq \frac{1}{12}.$0 under the same hypotheses; the $\langle Cov_K\,\xi,\xi\rangle \geq \frac{1}{12}.$1 factor could be removed under the much stronger condition that $\langle Cov_K\,\xi,\xi\rangle \geq \frac{1}{12}.$2 tiles $\langle Cov_K\,\xi,\xi\rangle \geq \frac{1}{12}.$3 by repeated reflections across its facets, which the paper does not pursue.

Limitations and open questions

Several qualifications are explicit in the paper. The claim that the Voronoi cell is always preferable among fundamental domains is asserted as plausible ("presumably") rather than proved. The Craig-lattice counterexample in Remark 5.2 is reported on ChatGPT's authority and explicitly not verified by the author. The sectional tiling criterion requires both a genuine hyperplane section that tiles and a variance concentration condition in the normal direction; whether some weaker structural condition suffices for the equivalence is not addressed. Finally, the equivalence between $\langle Cov_K\,\xi,\xi\rangle \geq \frac{1}{12}.$4 and $\langle Cov_K\,\xi,\xi\rangle \geq \frac{1}{12}.$5 under sectional tiling retains the $\langle Cov_K\,\xi,\xi\rangle \geq \frac{1}{12}.$6 gap inherited from the current state of the KLS problem, so closing it here is contingent on resolving KLS in full.

Conclusion

The paper distills a short chain of classical ideas — Haar-uniformity of fundamental domains, characters of the torus, Hadwiger's translation averaging, and log-concavity of marginals — into sharp spectral and isoperimetric lower bounds for flat tori in terms of covariance data of fundamental domains. Its principal contributions are the sharp $\langle Cov_K\,\xi,\xi\rangle \geq \frac{1}{12}.$7 variance constant, the resulting unconditional confirmation of the KLS conjecture for all flat tori (with dimension-free constants whenever the Voronoi cell is isotropic), the demonstration via Barnes–Wall and related lattices that no dimension-free upper bound on the spectral gap survives even under isotropy, and a clean sectional tiling criterion restoring two-sided equivalence. The note also transparently documents substantial use of a LLM in obtaining the sharp constant and the counterexamples, an increasingly relevant practice in mathematical research.