Restricted Hyperplane Sections of the Cross-Polytope and the Simplex

Abstract: We give a new proof of Webb's theorem on maximal central hyperplane sections of the regular (n)-simplex (Δ_n), viewed in its standard embedding in (\mathbb R^{n+1}). A similar method also yields sharp maximal estimates for non-central sections of (Δ^n) whose distance (d) from the barycenter is small, namely $d< \sqrt{\frac{1}{(n+1)(2n+1)}}.$ Moreover, we obtain sharp volume estimates for central hyperplane sections of the cross-polytope (B_1^n) that pass through the barycenter of a facet.

• The paper provides new proofs of classic extremal results for maximal central hyperplane sections of the simplex using Lagrange multipliers and second derivative analysis.
• It extends the analysis to near-central affine sections, revealing unique two-level vector configurations that determine maximal volumes.
• The study characterizes constrained cross-polytope sections via symmetric polynomial inequalities, linking geometric configurations to entropy and majorization theory.

Extremal Restricted Hyperplane Sections of the Cross-Polytope and the Simplex

Introduction and Context

The paper “Restricted Hyperplane Sections of the Cross-Polytope and the Simplex” (2606.07163) provides new structural and extremal results for the $(n-1)$-dimensional volume of hyperplane sections of the regular simplex $\Delta_n \subseteq \mathbb{R}^{n+1}$ and the cross-polytope $B_1^n \subseteq \mathbb{R}^n$, with special attention to sections obeying additional symmetry and barycentric constraints. The paper contributes new proofs of classical extremal theorems (notably Webb’s theorem for the simplex), sharp results for affine (non-central) sections with small deviation from the centroid, and a complete treatment of constrained sections of the cross-polytope through the barycenter of a facet. The work blends variational methods, Lagrange multiplier techniques, analysis of elementary symmetric polynomials, and Schur-convexity/majorization.

Simplex: Maximal Central Hyperplane Sections

The maximal volume problem for central hyperplane sections of the regular simplex $\Delta_n$ (standard embedding, centroid at $(1/(n+1),\dots,1/(n+1))$) has a long history. Webb [webb1996central] established that the maximal-volume $(n-1)$-section through the centroid is precisely when the hyperplane contains $n-1$ vertices, with the maximal value

$$\operatorname{vol}_{n-1}(\Delta_n \cap H_\text{max}) = \frac{\sqrt{n+1}}{(n-1)!} \frac{1}{\sqrt{2}}$$

The present paper provides a new proof of this extremum by analyzing the critical points of the mapping $$a \mapsto \operatorname{vol}_{n-1}(\Delta_n \cap a^\perp)$$ under the constraints $\sum a_j = 0$, $\Delta_n \subseteq \mathbb{R}^{n+1}$0, and showing all (local and hence global) maxima must have a unique negative coordinate, i.e., up to permutation, $\Delta_n \subseteq \mathbb{R}^{n+1}$1. The proof leverages Lagrange multipliers, explicit integral representations for the section volume (involving densities of exponential sums), and a careful second derivative analysis to exclude local maxima with more than one negative (or positive) coordinate. This structural theorem extends to all dimensions and does not use probabilistic or entropy methods used in prior literature—for instance, those in [tomasz].

The equality case and uniqueness (modulo permutation and sign) are established by reduction to a univariate optimization, with the volume formula interpretable in terms of densities of exponential random variable combinations. This also streamlines the route to confirming Webb’s spectral extremality without reliance on previously delicate probabilistic bounds.

Non-Central (Affine) Sections Near the Centroid

The authors extend the analysis to affine (non-central) sections of $\Delta_n \subseteq \mathbb{R}^{n+1}$2, characterized by a parameter $\Delta_n \subseteq \mathbb{R}^{n+1}$3 quantifying deviation from barycenter. By reparametrizing the hyperplane normal constraints, they show that for $\Delta_n \subseteq \mathbb{R}^{n+1}$4 (corresponding to affine sections at distance $\Delta_n \subseteq \mathbb{R}^{n+1}$5 from the barycenter), the maximal-volume affine section is determined by the explicit two-level vector configuration: $\Delta_n \subseteq \mathbb{R}^{n+1}$6 Classically, for symmetric convex bodies, extremal affine sections always contain the barycenter. For the simplex, this behavior is restricted and nontrivial due to asymmetry—yet, for sufficiently small $\Delta_n \subseteq \mathbb{R}^{n+1}$7 (i.e., close to central), the volume-maximizing section is proven to retain this two-level vector structure. The full argument involves a technical induction on the number of negative coordinates and uses the same Lagrange/second-order analysis adapted from the central case. The approach generalizes Webb's result and, importantly, characterizes the precise bifurcation in extremal structure as $\Delta_n \subseteq \mathbb{R}^{n+1}$8 increases.

Constrained Hyperplane Sections of the Cross-Polytope

For the cross-polytope $\Delta_n \subseteq \mathbb{R}^{n+1}$9 the authors study sections $B_1^n \subseteq \mathbb{R}^n$0 with two restrictions: the normal vector $B_1^n \subseteq \mathbb{R}^n$1 is unit length and $B_1^n \subseteq \mathbb{R}^n$2 (i.e., the hyperplane passes through the centroid of a facet). This restricted problem is fundamentally distinct from the symmetric (unrestricted) case.

A key technical tool is to express the section volume via a sum over elementary symmetric polynomials of $B_1^n \subseteq \mathbb{R}^n$3: $B_1^n \subseteq \mathbb{R}^n$4 This leads to analyzing extrema of $B_1^n \subseteq \mathbb{R}^n$5 over the constrained set, which is achieved by demonstrating that all maximizers and minimizers are two-level: all positive entries equal, all negative entries equal, possibly zeros. The authors provide explicit (and sharp) configuration results:

• The global minimum (for even $B_1^n \subseteq \mathbb{R}^n$6) is attained at the balanced half-plus/half-minus vector: $B_1^n \subseteq \mathbb{R}^n$7.
• For odd $B_1^n \subseteq \mathbb{R}^n$8, at a nearly balanced split with two levels, dictated by normalization and zero-sum constraint.
• They conjecture and provide compelling evidence that the global maximum, for small dimensions ($B_1^n \subseteq \mathbb{R}^n$9), coincides with the Webb-type vector (one positive, one negative, rest zeros), but for $\Delta_n$0 maximizers shift to the configuration conjectured to minimize the simplex section.

The method combines delicate analysis of the Schur-concavity of elementary symmetric functions, explicit construction/comparison of two-level vectors, and majorization relations. For all $\Delta_n$1, the maximal value is shown to occur at the most balanced two-level vector, with a monotonicity argument across possible zero-level groupings using majorization theory as developed in [marshall2011inequalities] and [bhatia1997matrix]. The $\Delta_n$2 case is distinguished and aligns with recent advances on the role of balanced configurations for $\Delta_n$3-norms and volume products of zero-sum vectors [zhang2026proof, holevo2026conjecture].

Connections and Auxiliary Results

The analysis establishes tight connections to entropy and log-concave probabilistic inequalities, via connections between the Fourier transform expressions for the section volume and Rényi entropies of mixtures of standard exponential or Laplace random variables. The tight symmetric polynomial inequalities derived for the main problem are of independent combinatorial interest and connect with Hunter’s positivity theorem and recent advances in sharp bounds for symmetric and power sum functions under majorization constraints [brazitikos2025sharp]. The general method and results also interface with classical problems in convex geometry such as the Mahler conjecture (see [karasev2021mahler]) and volume product optimization.

Implications and Further Directions

The structural results imply that in both the simplex and cross-polytope, symmetry-breaking under minimal additional constraints is highly restricted; two-level structures essentially dominate the extremal behavior. The methods employed are generic and adaptable, suggesting broad applicability to other regular polytopes and more general volume-product and entropy-type optimization problems.

On the theoretical front, the paper closes several open cases for affine sections near the centroid for the simplex and provides sharp answers for a new class of restricted problems for the cross-polytope. It leaves open the precise global maximizer for large $\Delta_n$4, where a possible phase transition appears.

Further directions include:

• Extending the two-level structural analysis to other families of convex bodies, such as $\Delta_n$5-balls for $\Delta_n$6.
• Demonstrating (or falsifying) the conjectured phase transition in maximizer structure as $\Delta_n$7 increases for cross-polytopes.
• Investigating minimal section volume configurations, especially for the simplex, where recent progress achieves improved bounds but exact minima remain elusive [tang2024simplex].
• Exploiting the polynomial bounds for symmetric functions in other contexts (e.g., operator theory, entropy inequalities, random matrix theory).

Conclusion

This work delivers a comprehensive variational and algebraic framework to address extremal restricted hyperplane section problems for the simplex and cross-polytope. The key advances are a streamlined proof of Webb's theorem, sharp classification of maximizer structure for near-central affine sections, and a resolution (including explicit formulas) of the minimal and maximal section volume configurations under barycentric facet constraints for the cross-polytope. The results are grounded in deep connections between convex geometry, majorization theory, and the analytic structure of symmetric polynomials, offering substantial new tools and insights for both geometric analysis and its probabilistic/entropy-theoretic avatars.