Bourgain’s Slicing Problem

• Bourgain’s Slicing Problem is a central question in convex geometry that asks if every unit-volume convex body in ℝⁿ has a hyperplane section with a universally bounded (n−1)-dimensional volume.
• Recent advances using stochastic localization, thin-shell estimates, and entropy methods have confirmed the conjecture by establishing a universal lower bound and refining bounds on the isotropic constant.
• The resolution connects deep concepts in isoperimetry, concentration of measure, and functional inequalities, paving the way for further research on general measures and symmetric convex bodies.

Bourgain’s Slicing Problem is a central open question in asymptotic convex geometry, pertaining to the minimal volume of central hyperplane sections of high-dimensional convex bodies. Formulated by Bourgain in the 1980s, it asks whether there exists a universal constant $c>0$ such that every convex body $K$ in $\mathbb{R}^n$ of volume one admits a hyperplane section of $(n-1)$-dimensional volume at least $c$, independent of the dimension. This conjecture, also known as the hyperplane conjecture, is equivalent to the boundedness of the so-called isotropic constant $L_K$ and underpins much of the structural theory of high-dimensional convex bodies, with deep connections to isoperimetry, concentration of measure, and functional inequalities. After three decades of technical development and partial results, Bourgain’s slicing problem was recently resolved in the affirmative, establishing the existence of a universal lower bound for maximal hyperplane sections of arbitrary convex bodies. The following exposition provides a rigorous account of the problem, including its generalizations, major approaches, key invariants, proof techniques, limitations for general measures, and its implications for related conjectures.

1. Formal Statement and Equivalence

Let $K\subset\mathbb{R}^n$ be a convex body of volume one, and let $H$ range over hyperplanes through the origin. Bourgain’s slicing problem asks whether

$\sup_{H} \mathrm{Vol}_{n-1}(K\cap H) \geq c > 0$

for a universal $c$, independent of $n$ and $K$. Equivalently, there exists a constant $C$ such that for every volume-one $K$,

$$L_K := |K|^{2/n}\cdot \det(\mathrm{Cov}(K))^{1/(2n)} \leq C,$$

where $L_K$ is the isotropic constant and $\mathrm{Cov}(K)$ is the covariance matrix of the uniform measure on $K$ (Klartag et al., 2024, Bizeul, 12 Jan 2025). The dual formulation, using hyperplane sections, is

$$|K| \leq C \cdot |K|^{1/n} \cdot \max_{\xi\in S^{n-1}} |K\cap\xi^{\perp}|.$$

2. Evolution and Partial Results

Initial quantitative estimates were polynomial in $n$: Bourgain proved $L_n=O(n^{1/4}\log n)$. Klartag (2006) improved this to $L_n=O(n^{1/4})$. Subsequent advances using stochastic localization, thin-shell analysis, and improved spectral inequalities drove the bound down from subpolynomial to

• $L_n=O(\exp[C\sqrt{\log n\log\log n}])$ (Chen),
• $L_n = O((\log n)^4)$ (Klartag–Lehec) (Klartag et al., 2022),
• $L_n = O(\log n)$ (Klartag, 2023) (Guan, 2024), and then $L_n = O(\log\log n)$ (Guan, 2024) (Guan, 2024).

Crucial milestones involved the links to the Kannan–Lovász–Simonovits (KLS) isoperimetric constant and the thin-shell constant $\sigma_n = \sup_\mu \mathrm{Var}_\mu(|x|^2)^{1/2}$ over isotropic log-concave measures. Conjecturally, bounding either the KLS constant or $\sigma_n$ would suffice to resolve the slicing conjecture (Klartag et al., 2022, Klartag, 2023).

The explicit reduction to centrally symmetric bodies (with optimal constant $1/2$) was established by Martín-Goñi (Martín-Goñi, 2022), effectively localizing difficulty to the symmetric setting.

3. Proof Techniques and Final Resolution

The affirmative solution is founded on three main analytic ingredients:

(a) Stochastic Localization:

Eldan’s process, later refined by Chen, Klartag, Lehec, Lee–Vempala, constructs a family of tilted log-concave densities

$$f_{t,\theta}(x) \propto \exp(\langle\theta,x\rangle - \frac{t}{2}|x|^2) f(x)$$

with $\theta_t$ driven by an SDE. The covariance process $A_t$ evolves according to

$dA_t = \text{(Itô correction)} - A_t^2\,dt,$

and tracking its moments controls concentration and section volume estimates (Klartag et al., 2022, Guan, 2024, Klartag et al., 2024, Bizeul, 12 Jan 2025).

(b) Thin-Shell/Small Ball Anti-Concentration:

Milman's $M$-ellipsoid theory and small-ball probability estimates—especially Paouris’ inequalities—bound the volume of $K$ intersected with Euclidean balls. The sharp recent improvement (Bizeul, (Bizeul, 12 Jan 2025)) establishes a small-ball exponent that is linear in $n$, eliminating the last logarithmic loss.

(c) Entropy and Shannon–Stam Stability:

Eldan–Mikulincer’s quantitative stability for the Shannon–Stam (entropy power) inequality, combined with de Bruijn’s identity, establishes control over the entropy deficit between isotropic log-concave measures and the Gaussian, bounding the isotropic constant by a universal value (Klartag et al., 2024).

Combined Outline:

• Begin with a volume-one convex body in isotropic position.
• Use stochastic localization to obtain, at deterministic or random time $t_0$, a log-concave measure with covariance trace comparable to $n$ (Klartag et al., 2024, Bizeul, 12 Jan 2025, Guan, 2024).
• Apply small-ball bounds and $M$-ellipsoid theory to show that $K$ cannot concentrate too much mass near the origin unless the isotropic constant is universally bounded.
• Use entropy arguments in combination with localization and stability to transfer the bound back to the original measure.

These constructions yield $\sup_n L_n < \infty$, resolving the slicing conjecture positively (Klartag et al., 2024, Bizeul, 12 Jan 2025).

4. Generalizations, Special Cases, and Limitations

Lower-Dimensional Sections and Arbitrary Measures

A principal generalization replaces hyperplanes with arbitrary $(n-k)$-dimensional affine sections and volume with even continuous measures $\mu$: $$\mu(L) \leq C^k\, |L|^{k/n}\, \max_{H\in\mathrm{Gr}_{n-k}} \mu(L\cap H)$$ for convex bodies $L\subset\mathbb{R}^n$ and $1\leq k < n$ (Koldobsky, 2014, Koldobsky et al., 2016, Klartag et al., 2017). In particular, Koldobsky proved strong positive results in special cases: unconditional convex bodies, duals with bounded volume ratio, $L_p$-balls with $p>2$, and the regime $k\geq\lambda n$. However, Klartag–Koldobsky (Klartag et al., 2017) constructed explicit examples showing that in the general measure setting, no dimension-free $C$ is possible: the best possible constant grows at least as $\sqrt{n}/\sqrt{\log\log n}$, and only special geometric (symmetry, volume ratio) or analytic (log-concavity, tail decay) constraints enable universal bounds. For $L_p$-balls with $p>2$, $C=\Theta(\sqrt{p})$ suffices (Koldobsky et al., 2016).

Reduction to Symmetric Bodies

Any convex body can be reduced to the centrally symmetric case at the cost of a best possible factor $1/2$ in the isotropic constant, with equality uniquely realized by simplices (Martín-Goñi, 2022). This reduction further clarifies that the extremal cases in the slicing conjecture are symmetric and highlights the geometric rigidity inherent in the problem.

Limits of Dimensional Dependence

For arbitrary densities, the measure slicing constant cannot be better than order $\sqrt{n}/\sqrt{\log\log n}$ even assuming tail decay, with only logarithmic improvements possible via improved analytic tools (Klartag et al., 2017). This suggests intrinsic limitations when moving beyond the uniform measure.

5. Connections to Other Geometric and Isoperimetric Problems

There are deep connections between the slicing constant $L_n$, thin-shell constant $\sigma_n = \sup_{\mu}\mathrm{Var}_\mu(|x|^2)^{1/2}$, and the KLS isoperimetric constant $U_n$. These invariants satisfy

$$L_n \leq C\sigma_n \leq C' U_n, \quad U_n\leq C''\sigma_n\log n,$$

and polylogarithmic bounds propagate between them via stochastic localization and spectral analysis (Klartag et al., 2022, Guan, 2024). Thus, advances in slicing directly inform isoperimetric and concentration inequalities (KLS conjecture, thin-shell, log-Sobolev).

A major application appears in covering questions such as Hadwiger’s conjecture: the improved slicing bound $L_n=O((\log n)^4)$ implies that every convex body in $\mathbb{R}^n$ can be covered by $\exp(-\Omega(n/(\log n)^8)) 4^n$ translates of its interior—an almost-exponential improvement over classical results (Campos et al., 2022).

6. Open Questions and Outlook

While Bourgain’s slicing problem is now affirmatively resolved up to universal constants, several quantitative and structural questions remain:

• Determination of the optimal constant $c$ in the volume lower bound for slices.
• Characterization of extremal bodies for which the bound is tight.
• Quantitative improvement of the constants via sharper control of the covariance process or entropy inequalities (Klartag et al., 2024).
• Full resolution in the general measure/section setting—beyond those classes where slicing holds with uniform constants.
• Further development of functional inequalities, stochastic processes, and symmetrization techniques to transfer these advances to other major conjectures (e.g., KLS).

7. Table: Milestones in the Slicing Constant $L_n$ Bounds

Author(s) and Year	Bound on $L_n$	Reference
Bourgain (1991)	$O(n^{1/4}\log n)$	(Guan, 2024)
Klartag (2006)	$O(n^{1/4})$	(Guan, 2024)
Chen (2021)	$< \exp[C(\log n)(\log\log n)]$	(Guan, 2024)
Klartag–Lehec (2022)	$O((\log n)^4)$	(Klartag et al., 2022)
Klartag (2023)	$O(\log n)$	(Guan, 2024)
Guan (2024)	$O(\log\log n)$	(Guan, 2024)
Klartag–Lehec & Bizeul (2025)	Universally bounded (solved)	(Klartag et al., 2024, Bizeul, 12 Jan 2025)

The resolution of Bourgain’s slicing problem concludes a foundational chapter in convex geometric analysis, with new avenues now open for the transfer of these tools to isoperimetric, spectral, and probabilistic questions in high dimensions.