Convex Geometric Foundations of Deviance

• Convex-geometric foundations for deviance are a unified framework that quantifies statistical extremality, spread, and rarity via convex analysis.
• They precisely characterize dominating points in Gaussian measures, derive tight variance bounds through simplex geometry, and employ valuation theory to define intrinsic metrics.
• This approach enables efficient large deviation analysis, optimal design under support constraints, and advanced risk measurement by leveraging duality and gauge functionals.

Convex-geometric foundations for deviance provide a unified geometric framework underlying the theory and quantification of statistical “deviance” across probability, statistics, and convex geometry. Deviance here refers to the quantification of rarity, spread, discrepancy, or extremality, with convex geometry providing sharp, structural characterizations of extremal points, rate functions, and optimal configurations. The foundational role of convexity permeates large deviation principles, moment inequalities, deviation measures, and geometric metrics, linking topics such as Gaussian extreme value statistics, optimal support-bounded measures, and valuation-based metrics on convex bodies.

1. Dominating Points and Large Deviations on Convex Sets

Central to the convex-geometric approach to large deviation analysis is the concept of the "dominating point" for Gaussian measures on convex sets. For extreme values of i.i.d. Gaussian random vectors $X_i \sim N(0,\Sigma)$, and for a closed convex set $C \subset \mathbb{R}^d$ (with nonempty interior and $0 \notin C$), the probability that the appropriately normalized block maximum $A_n^{-1}M_n$ falls into $C$ concentrates exponentially near a unique boundary point $x^*\in\partial C$. This point $x^*$ is determined as the solution to the convex quadratic program

$$x^* = \arg\min_{x \in C} \frac{1}{2} x^T \Sigma^{-1} x$$

and the associated rate function is $I_C(x^*) = \frac{1}{2} (x^*)^T \Sigma^{-1} x^*$ (Honnappa et al., 2018).

Geometrically, $x^*$ is the location where the first Gaussian level ellipsoid $C \subset \mathbb{R}^d$0 touches $C \subset \mathbb{R}^d$1. The "collusion" between the Gaussian covariance and the structure of $C \subset \mathbb{R}^d$2 dictates this point. In the polyhedral case $C \subset \mathbb{R}^d$3, the Karush-Kuhn-Tucker (KKT) conditions explicitly characterize $C \subset \mathbb{R}^d$4 in terms of the active constraints and Lagrange multipliers. This provides a precise convex-geometric mechanism by which rare events become exponentially concentrated near specific boundary points of $C \subset \mathbb{R}^d$5.

Extensions to Gaussian mixtures and importance sampling are also precisely governed by convex programs involving the geometry of $C \subset \mathbb{R}^d$6 and the parameters of the underlying distributions, with the dominating point serving as the mode of the optimally-tilted measure (Honnappa et al., 2018).

2. Variance Bounds, Support Geometry, and Symmetry Breaking

Variance and more general recentered moments under support constraints are governed by convex-geometric extremal principles. For a random vector $C \subset \mathbb{R}^d$7 supported in a compact set $C \subset \mathbb{R}^d$8, the variance $C \subset \mathbb{R}^d$9 is maximized (under the constraint $0 \notin C$0) by placing mass symmetrically at the vertices of a regular $0 \notin C$1-simplex of diameter $0 \notin C$2. This is a higher-dimensional generalization and sharpens one-dimensional inequalities of Popoviciu and Bhatia-Davis (Lim et al., 2020).

The maximal variance is

$0 \notin C$3

with equality only for uniform measures on the simplex. Convex-duality methods, specifically infinite-dimensional linear programming and Legendre–Fenchel theory, provide the route to these tight bounds. The geometry of the convex hull, minimal enclosing balls, and Jung’s theorem identify and guarantee the extremal configuration. The result reveals a symmetry-breaking phenomenon in which simplex-supported measures, rather than diffuse distributions (e.g., spheres), maximize variance for fixed diameter (Lim et al., 2020).

3. Valuation-Based Metrics and Intrinsic Deviance

Valuation theory offers a structural framework for measuring deviance between convex bodies $0 \notin C$4 via deviations associated with intrinsic volumes or more general valuations. For a translation-invariant, continuous valuation $0 \notin C$5, two canonical deviations are defined:

• Meet deviation: $0 \notin C$6,
• Join deviation: $0 \notin C$7, where $0 \notin C$8 is the convex hull of the union.

Under monotonicity, these are nonnegative and become semimetrics. For intrinsic volumes $0 \notin C$9, these recover and interpolate classical metrics such as the symmetric difference (for volume) and $A_n^{-1}M_n$0-distance of support functions (for mean width). The intrinsic metric construction ($A_n^{-1}M_n$1) restores triangularity by considering piecewise-metric-continuous paths, connecting these deviations to genuine metric geometry (Cutler et al., 28 Nov 2025).

The duality between $A_n^{-1}M_n$2 and $A_n^{-1}M_n$3 is precise: for $A_n^{-1}M_n$4-homogeneous (mean width) and $A_n^{-1}M_n$5-homogeneous (volume) valuations, these deviations reduce to known geometric distances, and lengths of Minkowski interpolations give shortest paths. The resulting structure elucidates the geometric content of deviation beyond the Hausdorff metric and opens new avenues for statistical shape analysis, geometric inference, and convex body approximation (Cutler et al., 28 Nov 2025).

4. Gauge Functionals and Deviation Measures

A unified convex-geometric theory of deviation measures arises from the Minkowski (gauge) functional associated to convex “acceptance sets.” Given a subset $A_n^{-1}M_n$6 in a vector space (or space of random variables), the gauge

$A_n^{-1}M_n$7

defines a generalized deviation measure provided $A_n^{-1}M_n$8 is star-shaped at $A_n^{-1}M_n$9, stable under addition of constants, convex, and radially bounded at non-constants. This construction recovers all classical positive-homogeneous deviation measures (e.g., mean absolute deviation, standard deviation, semi-deviation, CVaR-deviation) as gauges of their appropriate acceptance sets (Moresco et al., 2020).

Convexity of $C$0 yields convexity of $C$1; stability under additive constants yields translation-insensitivity. Duality theory (Fenchel–Moreau) links $C$2 to support functionals of the polar set $C$3, and properties of $C$4 correspond bijectively to properties of the associated deviation measure. For variance, the gauge of its $C$5-sublevel set yields the standard deviation. This perspective reveals that the spread or deviance is essentially a geometric “distance from acceptability,” measured by contraction to the acceptance cone (Moresco et al., 2020).

5. Convex-Geometric Large Deviations for Mixed and Point Processes

Beyond classical LDPs, convex-geometric methods produce sharp large deviation scaling for both mixed measures and spatial point processes. For mixed measures arising from log-concave densities driven by convex gauges, the first-order mixed measure $C$6 is controlled by the $C$7-inradius of $C$8: $C$9 where $x^*\in\partial C$0 (Lafi et al., 21 Feb 2026). This result holds uniformly across log-concave densities, generalizing the role of the gauge body in determining exponential decay rates for rare geometric events.

In the combinatorial setting, the convex distance of Talagrand provides a dimension-free, non-asymptotic metric for large deviations in both product spaces and random point processes (e.g., binomial or Poisson). Given measurable event sets $x^*\in\partial C$1 in the Poisson process configuration space, the convex distance $x^*\in\partial C$2 yields the bound: $x^*\in\partial C$3 where $x^*\in\partial C$4, thus providing a convex-geometric underpinning for rare-event probabilities in random geometric and combinatorial structures (Reitzner, 2013).

6. Synthesis: Unifying Principles and Further Directions

The convex-geometric approach is systematic across deviance notions: extremal points (dominating points) for large deviations, moment bounds controlled by convex hull geometry, deviation metrics from valuation differences, and shrinkage factors arising from gauge functionals. All constructions embody a precise correspondence between geometric configuration—convex set, support, mixed body, or acceptance region—and the extremality or rarity of the associated probabilistic or statistical event.

Applications span rare-event simulation (where tilting via the dominating point yields asymptotically efficient algorithms), optimal design in support-constrained statistics, geometric shape comparison, and optimization under convex constraints. Open directions include characterizing all valuations whose associated deviations are genuine metrics; extending gauge-based constructions to non-homogeneous settings; and sharpening quantitative comparison theorems in geometric probability. The geometric foundation underpins and often sharpens classical analytic inequalities, revealing the optimality mechanisms hidden in convex structure.

The convex-geometric foundations for deviance thus form an integrative paradigm, with deep ramifications for the understanding and computation of statistical extremality, spread, and discrepancy, governed by the geometry of convex bodies, duality, and optimization (Honnappa et al., 2018, Lim et al., 2020, Cutler et al., 28 Nov 2025, Lafi et al., 21 Feb 2026, Moresco et al., 2020, Reitzner, 2013).