Aomoto interpolation and Coxeter systems

Technical Summary: Aomoto Interpolation and Coxeter Systems

Problem Statement
The paper addresses two interconnected problems in geometric analysis and probability theory: the characterization of extremal configurations for the strong polarization inequality and the proof of a generalized Gaussian Product Inequality (GPI).

1. Strong Polarization Inequality: The authors investigate the inequality $\sum_{j=1}^n \langle v_j, u \rangle^{-2} \leq n^2$ for unit vectors $v_1, \dots, v_n \in S^{d-1}$. While the bound is known to be sharp, the specific geometric structures (extremal configurations) that attain this bound for all directions $u$ were not fully characterized. Previous work established the inequality but left the rigidity of the equality case open.
2. Gaussian Product Inequality: The paper seeks to generalize the GPI, recently proved by Ouimet and Greaves, to a broader class of functions (completely monotone functions) and arbitrary hyperplane arrangements, avoiding reliance on specific correlation matrix computations or special function identities.

Methodology
The core methodology relies on the construction of a Lagrange-type basis for the Aomoto space $AO(\mathcal{A})$, a vector space of rational functions associated with a central hyperplane arrangement $\mathcal{A} = \{v_1^\perp, \dots, v_n^\perp\}$.

• Aomoto Space and Dimension: The authors utilize the Orlik-Terao dimension theorem, which states that $\dim AO(\mathcal{A})$ equals the number of chambers (connected components) of the arrangement.
• Chamber Basis Construction: For each local extremum $u$ of the polynomial $P(x) = \prod_{j=1}^n \langle v_j, x \rangle$ on the unit sphere (which corresponds one-to-one with the chambers), the authors define a rational function $L_u(x)$. This function is constructed using a positive weight $\mu(u)$ derived from a determinant involving the vectors $v_j$ and the extremum $u$.
• Interpolation Formula: The paper proves that $\{L_u\}_{u \in E(P)}$ forms a basis for $AO(\mathcal{A})$. Consequently, any function $f \in AO(\mathcal{A})$ satisfies the interpolation formula:
  $$f(x) = \sum_{u \in E(P)} f(u) L_u(x)$$
  This formula allows the translation of global identities in the space of rational functions into finite sums over the critical points (extrema).
• Generalized Setting: To handle the Gaussian inequalities, the authors introduce a generalized setting with parameters $t_j$ and a shifted map $T(x)$, establishing a "chamberwise change of variables" that relates integration against a modified measure to a weighted sum over the extremal points.

Key Contributions and Results

1. Interpolation Theorem (Theorem A): The paper establishes that the functions $L_u$ form a basis for the Aomoto space and satisfy the Lagrange property $L_u(w) = \delta_{u,w}$. This yields an explicit interpolation formula for elements of $AO(\mathcal{A})$.
2. Chamber Identities (Theorems 2.4 & 2.5): By applying the interpolation formula to specific functions (constants, reciprocals, and quadratic forms), the authors derive a family of "chamber identities" for the weights $\mu(u)$. These include:
   • $\sum \mu(u) = 1$
   • $\sum \mu(u) \langle v_i, u \rangle^{-1} = 0$
   • $\sum \mu(u) \langle v_i, u \rangle^{-1} \langle v_j, u \rangle^{-1} = n \delta_{ij}$
   • A crucial identity involving the sum of squared reciprocals: $\sum \mu(u) (\sum \langle v_j, u \rangle^{-2} - n^2) = 0$.
3. Characterization of Extremizers (Theorems B & C):
   • Using the chamber identities, the authors prove that if a configuration $v_1, \dots, v_n$ is extremal (attaining the bound $n^2$ for all $u$), the product polynomial $P(x)$ must be harmonic ($\Delta P = 0$).
   • Combined with Agranovsky's Theorem and previous results, this leads to Theorem C: The equality case holds if and only if the set of vectors $\Phi = \{\pm v_1, \dots, \pm v_n\}$ forms a finite reflection system (a finite Coxeter group orbit). This identifies orthonormal bases and regular planar configurations as the simplest members of a much larger family governed by Coxeter symmetry.
4. Generalized Gaussian Product Inequality (Theorem D):
   • The authors prove that for unit vectors $v_j$ and completely monotone functions $f_j$, the inequality $E[\prod f_j(\langle v_j, Z \rangle^{-2})] \geq \prod E[f_j(Z^{-2})]$ holds.
   • The proof adapts the recent breakthrough of Ouimet and Greaves by utilizing the Aomoto space analysis and the generalized chamber identities (Lemma 4.3). This approach avoids direct computation with the correlation matrix, instead relying on the change of variables $T(x)$ and Jensen's inequality applied to a specific probability measure constructed from the chamber weights.

Significance and Claims
The paper claims to uncover and generalize the algebraic structure underlying the mechanism used in recent proofs of the Gaussian Product Inequality and polarization problems.

• Unification: The interpolation theorem unifies the derivation of algebraic identities previously obtained via the Euler-Jacobi vanishing theorem (for general arrangements) and the specific determinant identities used in the Gaussian proof (which required maximal chamber counts). The authors note that their approach applies to arbitrary central hyperplane arrangements, including those with linear dependencies among defining vectors.
• Rigidity: The paper provides a complete characterization of the equality cases for the strong polarization inequality, confirming that they are precisely the finite reflection systems.
• Generalization: The methodology extends the Gaussian Product Inequality to completely monotone functions and arbitrary arrangements, offering a proof that is structurally distinct from previous approaches by isolating the general principle (interpolation and change of variables) rather than relying on special function properties.

The authors explicitly state that their derivation of the algebraic identities is not independent of the recent work by Ouimet and Greaves but rather offers a new perspective that generalizes the underlying mechanism to broader geometric settings.