Gaussian Measures of Inner Cones

• Gaussian measures of inner cones quantify the Gaussian mass within convex subregions using conic intrinsic volumes and solid angles.
• Key methodologies involve analyzing intrinsic volumes, gauge functions, and phase transition thresholds to inform high-dimensional optimization problems.
• Applications span harmonic analysis, stochastic geometry, and convex optimization, enabling precise error bounds and variational solutions.

The Gaussian measure of inner cones refers to the probabilistic and geometric quantification of subsets within convex cones—typically those “internal” or “admissible” with respect to either a base point, a direction, or an asymptotic property—with respect to the standard Gaussian (radial log-concave) measure. This concept appears across harmonic analysis, stochastic geometry, convex optimization, geometric measure theory, and high-dimensional probability. The following sections delineate its definitions, analytic structures, geometric implications, role in inverse problems and optimization, and connections to Minkowski-type problems in the Gaussian setting.

1. Foundational Definitions and Analytical Structures

In $\mathbb{R}^n$, the Gaussian measure $\gamma$ is defined as

$$d\gamma(x) = (2\pi)^{-n/2} e^{-\frac{|x|^2}{2}} dx.$$

The measure of a cone $C \subset \mathbb{R}^n$ consists of the Gaussian “mass” lying within $C$, typically written as $\gamma(C) = \mathbb{P}(g \in C)$ for $g \sim N(0,I_n)$. Solid angular (“conic”) measures are the archetypal example:

$\alpha(C) := \gamma(C).$

In advanced contexts, the “inner cone” terminology specializes to admissible or local cones, as in the definitions of non-tangential (admissible) cones $\Gamma_a(x)$ in harmonic analysis (Maas et al., 2010) or the directional cones associated with the geometry of $C$-pseudo-cones (Wang et al., 30 Dec 2024, Shan et al., 27 Jan 2025, Shan et al., 2 Mar 2025). The integral geometry of convex cones parameterizes interior structures via intrinsic volumes $v_k(C)$, solid angles, and projection relations—all of which are expressible through the Gaussian measure.

For unbounded convex sets (C-pseudo-cones), inner cones relate to the asymptotic structure at infinity governed by a recession cone $C$. Here, the Gaussian surface area measure is weighted by $e^{-\frac{|x|^2}{2}}$ over the boundary “facing” the inner cone:

$$S_\gamma(K, \omega) = (2\pi)^{-n/2} \int_{\nu_K^{-1}(\omega)} e^{-\frac{|x|^2}{2}} d\mathcal{H}^{n-1}(x)$$

for Borel $$\omega \subset \mathbb{S}^{n-1}\cap \text{int}\,C^\circ$$ (Wang et al., 30 Dec 2024, Shan et al., 27 Jan 2025).

2. Geometric Quantities: Conic Intrinsic Volumes, Solid Angles, and Gauge Functions

The mathematical characterization of the measure of inner cones leverages several key geometric quantities:

• Conic Intrinsic Volumes: For a convex cone $C \subset \mathbb{R}^n$, the sequence $v_k(C)$ ($k=0,\ldots,n$) gives a probability measure on $\{0,\ldots,n\}$. The master Steiner formula states that the distribution of the squared length of the projection of $g\sim N(0,I_n)$ onto $C$ is a mixture of $\chi^2$ distributions weighted by $v_k(C)$ (McCoy et al., 2013, Goldstein et al., 2014):

$$\mathbb{P}(|\Pi_C(g)|^2 \in A) = \sum_{k=0}^n v_k(C) \mathbb{P}(Y_k \in A)$$

where $Y_k \sim \chi^2_k$.

• Solid Angles: The solid angle of $C$, $\alpha(C)=\gamma(C)$, quantifies the Gaussian measure of the cone. For polyhedral cones associated to regular polytopes (simplices, cubes, crosspolytopes), explicit formulas for solid angles and intrinsic volumes reduce the computation of various stochastic geometric quantities to sums over cones (Kabluchko et al., 2020).
• Gauge Functions: In the context of normed linear spaces, the gauge (or Minkowski) function $p(x)$ associated to a cone $K$ is sublinear and satisfies $p(x)<0$ if and only if $x$ is in the interior of $-K$. The Gaussian measure of the inner cone (interior of $-K$) is $\gamma(\{x: p(x) < 0\})$, linking function analysis to measure geometry (Svaiter, 2011).

3. Phase Transitions, Concentration, and Berry–Esseen Bounds

So-called “Gaussian measure of inner cones” is deeply intertwined with phase transitions and concentration phenomena in high-dimensional probability and convex optimization:

• Phase Transitions: In linear inverse problems with convex constraints, the existence of solutions (e.g., in compressed sensing or cone-constrained logistic regression) is governed by sharp thresholds involving the statistical dimension $\delta(K) = \mathbb{E}\|\Pi_K(g)\|^2$ of a cone $K$:

$$\mathbb{P}(L \cap G K \neq \{0\}) \approx \begin{cases} 1 & \sqrt{\delta(K)} > \sqrt{m - \delta(L)} + C\sqrt{t} \ 0 & \sqrt{\delta(K)} < \sqrt{m - \delta(L)} - C\sqrt{t} \end{cases}$$

where $G$ is a Gaussian random matrix and $L$ is a fixed cone (Han et al., 2022).

• Concentration of Intrinsic Volumes: For high-dimensional cones, the random variable $V_C$ (index distributed by conic intrinsic volumes) satisfies central limit theorems, and explicit Berry–Esseen-type bounds (Goldstein et al., 2014):

$$\sup_{u\in \mathbb{R}} \left| \mathbb{P}\left( \frac{V_C-\delta_C}{\sqrt{T_C}} \leq u \right) - \Phi(u) \right| = O\left(\frac{1}{\log \delta_C}\right)$$

where $T_C$ is the variance of $V_C$.

• Exact Regularization of LPs: In regularized linear programming, the feasibility and stability of solutions under perturbations are governed by the Gaussian measure of “shifted inner cones”:

$$\gamma(N(z) \cap [N(z) + \partial(\epsilon\psi)(z)]) \geq \gamma(N(z)) \exp\left(-\frac{1}{2}\epsilon^2 \|\mathbf{v}_z\|^2 - \epsilon \|\mathbf{v}_z\| \sqrt{n}\right)$$

where $N(z)$ is the normal cone at vertex $z$ and $\epsilon$ the regularization strength (Friedlander et al., 15 Oct 2025).

4. Variational and Minkowski-Type Problems in the Gaussian Setting

A growing body of work develops a Minkowski-type theory adapted to unbounded sets (C-pseudo-cones) and weighted by the Gaussian density:

• Gaussian Minkowski Problem: Given a finite Borel measure $\mu$ on $\omega \subset S^{n-1}\cap\text{int } C^\circ$, does there exist a C-pseudo-cone $K$ such that the (normalized) Gaussian surface area measure $S_\gamma(K, \cdot)$ equals $\mu$ (or up to a constant)? Existence and uniqueness can be established via variational methods for small co-volume (measured by $\gamma_n(C\setminus K)$), with uniqueness often conditional on fixed co-volume constraints (Wang et al., 30 Dec 2024, Shan et al., 27 Jan 2025).
• L$_p$ Gaussian Minkowski Problem: The extension to $L_p$ surface area measures, incorporating support function and exponent $p$, yields existence and (sometimes) uniqueness for a wide range of $p$, further linking the geometry of inner cones and prescribed Gaussian measures (Shan et al., 2 Mar 2025):

$$S_{p, \gamma}(K, \omega) = (2\pi)^{-n/2} \int_{\nu^{-1}_K(\omega)} |h_K(x)|^{1-p} e^{-|x|^2/2} d\mathcal{H}^{n-1}(x)$$

• Weighted Isoperimetry in Cones: Weighted isoperimetric inequalities in cones with Gaussian-like densities $d\mu=a x_N^k e^{c|x|^2}dx$ connect isoperimetric minimizers to geometric structure and spectral inequalities; separating radial and angular components enables a precise characterization of which cones and densities admit ball–cap intersecting minimizers (and, hence, which inner cones possess optimal Gaussian measure properties) (Brock et al., 2011).

5. Applications in Harmonic Analysis, Stochastic Geometry, and Optimization

The Gaussian measure of inner cones has a pivotal role across analytic, geometric, and algorithmic contexts:

• Harmonic Analysis: In Gaussian Hardy spaces, the analysis of non-tangential maximal and conical square functions requires working with admissible cones defined via $m(x) = \min\{1, 1/|x|\}$ to exploit local doubling properties. The measure of these cones under $\gamma$ enables the transfer of mapping properties and endpoint results as in the Euclidean setting (Maas et al., 2010).
• Stochastic Geometry: Absorption probabilities, face counts, and random sections/projections of polytopes reduce to explicit sums or integrals of Gaussian measures over inner cones associated with polyhedral geometry, utilizing solid angles and conic intrinsic volumes (Kabluchko et al., 2020).
• Convex Optimization and Statistical Inference: The geometry of descent cones, normal fans, and shifted cones controls recovery thresholds, minimax testing radii, and regularization thresholds via their Gaussian measure. Results are notably sharp in high-dimensional regimes due to concentration properties and facilitate precise error bounds in generalized likelihood ratio testing (Wei et al., 2017, Goldstein et al., 2014, Friedlander et al., 15 Oct 2025).

6. Extensions: Metric Geometry and Invariance Properties

Beyond real vector spaces, the metric geometry of inner cones has been extended to spaces of positive-definite (and semi-definite) matrices via Hilbert geometry. For the symmetric positive-definite bicone—relevant to extended Gaussian families with degenerate covariances or precisions—the Hilbert metric is

$$d_H(A,B) = \log \left( \frac{\max\{ \lambda_{\max}(B^{-1}A), \lambda_{\max}((I - B)^{-1}(I - A)) \}}{\min\{ \lambda_{\min}(B^{-1}A), \lambda_{\min}((I - B)^{-1}(I - A)) \}} \right)$$

and is invariant under complement and orthogonal conjugation (Karwowski et al., 20 Aug 2025). This metric structure supports the paper of Gaussian measures in degenerate or extended parameter spaces, mapping their geometric “inner cones” to spectral invariants.

7. Synthesis and Outlook

The Gaussian measure of inner cones is the fundamental probabilistic-geometric tool connecting the concrete “size” or “mass” of subregions of cones (in Euclidean, Hilbert, matrix, or abstract spaces) to probabilistic, analytic, and algorithmic outcomes. It encodes phase transition thresholds, facilitates explicit variational formulations for existence and uniqueness theorems in Gaussian Minkowski-type problems, and yields dimension-dependent scaling laws for the robustness of solutions in statistical and optimization models. The continued development of this concept—especially in relation to variational problems, intrinsic volumes, and metric invariants—will likely spur further advances in geometric analysis, high-dimensional probability, and the mathematics of uncertainty and optimization.