Cone-Restricted Rayleigh Inequalities

• Cone-restricted Rayleigh inequalities are spectral and convexity inequalities defined on proper convex cones, generalizing classical Rayleigh-quotient approaches under conic constraints.
• They incorporate log‐concavity and negative dependence via the Rayleigh matrix, offering multidirectional bounds and stability guarantees for eigenfunctions and probability measures.
• These inequalities bridge variational analysis, isoperimetry, and harmonic analysis, with concrete applications in conic optimization and Lyapunov stability of dynamical systems.

Cone-restricted Rayleigh inequalities are spectral, convexity, and negative-dependence inequalities for polynomials, eigenfunctions, or measures, formulated with explicit dependence on an ambient proper convex cone constraints. They generalize classical Rayleigh—quotient type inequalities to settings where positivity, log-concavity, or variance bounds are required only within a specified cone, unifying approaches from polynomial invariants, variational analysis, isoperimetry, and harmonic analysis on conic domains. These inequalities play a central role in the structure theory of Lorentzian and log-concave polynomials, spectral optimization under conic constraints, and discrete probability, with connections to barrier methods for conic optimization and Lyapunov stability in cone-constrained dynamical systems.

1. Proper Convex Cones, Lorentzian Forms, and Cone-associated Structures

A proper convex cone 𝐾⊆ℝⁿ is a closed, convex set with nonempty interior, containing no nontrivial subspace. The dual cone is 𝐾* = { y∈ℝⁿ: yᵀx≥0 ∀x∈𝐾 }. A homogeneous polynomial f∈ℝ[x₁,…,xₙ] of degree d≥2 is called 𝐾–Lorentzian if, for all sequences of (d–2) directions a₁,…,a_{d–2} in int 𝐾, the (d–2)-fold directional derivative is a quadratic form Q with exactly one positive eigenvalue, satisfying ⟨y,Qx⟩>0 for all x,y∈int 𝐾. This is equivalent to requiring all lower-order mixed directional derivatives are log-concave on int 𝐾 ("𝐾–completely log-concave").

Given f and v∈int 𝐾, associate:

• The open cone K°(f,v) = { x∈ℝⁿ: f(x)>0, D_v f(x)>0, ..., D_v^{d–1}f(x)>0 }
• The closed cone K(f,v) = { x∈ℝⁿ: f(x)≥0, D_v f(x)≥0, ..., D_v^{d–1}f(x)≥0 }

One has 𝐾 ⊆ K(f,v) ; additionally, if f is Lorentzian on K(f,v), then K(f,v) is convex, generalizing hyperbolicity cones to the Lorentzian setting (Dey, 24 Dec 2025).

2. The Rayleigh Matrix and Log-Concavity

Let f be twice continuously differentiable in a neighborhood of int 𝐾. The Rayleigh matrix associated to f at x is the symmetric matrix:

$$M_f(x) \;:=\; \nabla f(x)\,\nabla f(x)^T \;-\; f(x)\,\nabla^2 f(x)$$

A direct computation shows the Hessian of log f,

$$\nabla^2\log f(x) = \frac{1}{f(x)} \nabla^2 f(x) - \frac{1}{f(x)^2}\nabla f(x)\nabla f(x)^T,$$

so

$- \nabla^2\log f(x) = \frac{1}{f(x)^2} M_f(x)$

Therefore, M_f(x)≽0 if and only if log f is concave at x. The matrix M_f(x) provides a multidirectional refinement of log-concavity and negative-dependence, and its spectral properties are tightly coupled to conic constraints (Dey, 24 Dec 2025).

3. Single- and Two-Direction Rayleigh Inequalities

Scalar (Single-Direction) Rayleigh Inequalities

For f 𝐾–Lorentzian, x∈𝐾, and any u∈ℝⁿ:

$$R_u f(x) := (D_u f(x))^2 - f(x)\, D_u^2 f(x) = u^\top M_f(x) u \ge 0$$

This expresses the nonnegativity of the Rayleigh quadratic form along cone directions, a direct extension of the classical Rayleigh difference to cone-restricted polynomials (Dey, 24 Dec 2025).

Mixed (Two-Direction) Rayleigh Inequalities and Acuteness

Define the mixed Rayleigh difference:

$$R_{v,w}f(x) := D_v f(x) D_w f(x) - f(x) D_v D_w f(x) = v^\top M_f(x) w$$

For x∈int 𝐾, the following are equivalent:

1. $R_{v,w} f(x)\ge0$ for all v,w∈𝐾.
2. $v^\top M_f(x)w\ge0$ for all v,w∈𝐾.
3. 𝐾 is "acute" with respect to M_f(x), i.e. the associated bilinear form $\langle v, w\rangle_{M_f(x)}$ is nonnegative for all v,w∈𝐾.

If 𝐾 = cone{u₁,…,uₘ}, verification only on generator pairs (u_i, u_j) is sufficient (Dey, 24 Dec 2025).

4. Probabilistic, Spectral, and Negative Dependence Interpretations

Suppose f(x)=∑{α∈ℕⁿ} cα x^α, c_α≥0, x>0, is a partition-function style polynomial. With the associated Gibbs measure μx(α) = cα x^α/f(x), in log-coordinates the Hessian of log f gives covariances:

$$\frac{\partial^2}{\partial \theta_i \partial\theta_j} \log f(e^\theta) = Cov_{\mu_x}(α_i, α_j)$$

Thus,

$M_f(x) = -f(x)^2 \nabla^2 \log f(x)$

is (up to scale) the negative covariance operator. The scalar Rayleigh difference along coordinate axes reduces to:

$Δ_{ij}f(x) = x_i x_j Cov_{\mu_x}(α_i, α_j)$

Cone-restricted Rayleigh inequalities impose that, for any u,v∈𝐾, Cov_{\mu_x}(u·α, v·α)≤0, extending strongly Rayleigh/negative association theory to cone-compatible directions (Dey, 24 Dec 2025).

This framework connects to fast mixing of Glauber dynamics, concentration via Brascamp–Lieb and Bakry–Émery criteria, and spectral independence phenomena in probability.

5. Concrete Examples: Determinantal Polynomials and Semipositive Cones

For A∈ℝ^{n×n} nonsingular, set diagonal slices D_j=Diag(a_{1j},…,a_{nj}) and

$f_A(x) = \det\left(\sum_{j=1}^n x_j D_j\right)$

A direct computation yields:

• The hyperbolicity cone: $$\Lambda_+(f_A,e) = \{x\in\mathbb{R}^n: \sum_j x_j D_j \succeq 0\} = \{x: Ax\geq 0\}$$
• Intersecting with the nonnegative orthant recovers the classical semipositive cone: $K_A = \{x\geq 0: Ax\geq 0\}$

On K_A, f_A is K_A–Lorentzian, and all cone-restricted Rayleigh inequalities hold. In particular:

$$(D_u f_A(x))^2 - f_A(x) D_u^2 f_A(x) \geq 0,\quad \forall u\in K_A$$

$$R_{v,w} f_A(x) = v^\top M_{f_A}(x) w \geq 0, \quad \forall v,w\in K_A$$

This provides a hyperbolic/Lorentzian barrier for conic optimization over K_A and establishes a family of negatively-dependent Gibbs measures in the determinantal regime (Dey, 24 Dec 2025).

6. Cone-Restricted Rayleigh Inequalities in Harmonic and Spectral Analysis

In the setting of conic domains for eigenvalue problems and isoperimetry, the Rayleigh quotient admits explicit lower bounds governed by cone geometry and weight:

$$\lambda_1(W;B) \ge N-1 + r^2 \frac{(A'(r))^2}{A(r)^2} - r \frac{A'(r)}{A(r)} - r^2 \frac{A''(r)}{A(r)}, \quad \forall\,r>0$$

for the first nonzero Neumann eigenvalue on conic spherical slices, with A(r) and B(θ) as factorized weight components (Brock et al., 2011, Berchio et al., 12 Apr 2025). In sharp form, with Dirichlet Laplace–Beltrami spectral parameter λ₁(Σ),

$$\int_{C_Σ} |\nabla u|^2 dx \ge \lambda_1(\Sigma) \int_{C_Σ} |u|^2 dx$$

This matches the Poincaré inequality, and all constants are sharp in this cone-restricted framework (Berchio et al., 12 Apr 2025). The approach blends Emden–Fowler transforms and separation of variables for reduction to one-dimensional Hardy and Rellich inequalities.

7. Connections, Generalizations, and Applications

Cone-restricted Rayleigh inequalities provide a robust framework for:

• Certifying log-concavity and negative dependence within conic domains
• Establishing Lyapunov stability for EVI/LEVI systems with cone-invariant quadratic forms: if q(x)=x^T A x is (strictly) 𝐾–Lorentzian, then A is (strictly) 𝐾–copositive and yields Lyapunov (semi-)stability on 𝐾
• Supporting hyperbolic barrier methods for interior-point conic optimization
• Developing spectral bounds for Neumann Laplacians and weighted inequalities on conic domains
• Producing sharp decoupling and restriction estimates for the Fourier extension operator to the truncated cone, including small-cap decoupling, in harmonic analysis (Maldague et al., 2022)

These inequalities generalize classical isoperimetric, Poincaré, and log-concavity statements to a cone-adapted setting, providing maximal domains for spectral and stochastic optimality (Dey, 24 Dec 2025).