Maximum Volume Criterion

• The Maximum Volume Criterion is a mathematical framework that establishes upper bounds for volumes in geometric, algebraic, and physical systems under fixed constraints.
• It underpins rigidity results in Riemannian and Alexandrov geometry, facilitates robust matrix approximations, and guides experimental design via submatrix selection.
• Its applications span complex geometry, hyperbolic models, variational metrics, and physical systems such as wave energy devices and droplet interactions.

The Maximum Volume Criterion encompasses a broad set of mathematical results and methodologies that determine upper bounds—or in some cases, characterize extremal attainments—for the volume of geometric objects, algebraic structures, or physical configurations subject to fixed constraints. Applications span Riemannian and Alexandrov geometry, combinatorial and matrix analysis, mathematical physics, and optimization in both pure and applied settings. This article synthesizes the principal formulations, rigidity results, algorithms, and implications associated with maximum volume phenomena, with precise citation to foundational work.

1. Geometric and Topological Maximum Volume Rigidity

A recurring theme in global Riemannian and Alexandrov geometry is that among all spaces of prescribed curvature and diameter or radius, certain model spaces uniquely attain the maximal possible volume. The canonical result, derived from Bishop–Gromov comparison and further developed in the context of rigidity, asserts that for closed Riemannian $n$-manifolds $(M^n, g)$ with sectional curvature $\geq k$ and radius at most $r$,

$\operatorname{Vol}(M) \leq V_k(r),$

where $V_k(r)$ denotes the volume of the ball of radius $r$ in the constant curvature space form $M_k^n$ (Pro et al., 2012). Near equality in this bound leads to diffeomorphism rigidity: as shown in "The Diffeomorphism Type of Manifolds with Almost Maximal Volume", if $\operatorname{Vol}(M)$ is within $\varepsilon_0(n, k, r)$ of $V_k(r)$, then $M$ is diffeomorphic to $S^n$ or $\mathbb{R}P^n$.

Within Alexandrov geometry, Li–Rong implement a "relative maximum volume" principle for non-smooth spaces, showing that if $f: Z \to X$ is a distance non-increasing, onto map between compact Alexandrov $n$-spaces with curvature $\geq \kappa$ and $\operatorname{vol}(Z)=\operatorname{vol}(X)$, then $X$ is isometric to a quotient of $Z$ along the boundary (Li et al., 2011). The "Pointed Bishop–Gromov" version provides rigidity at every scale, forcing local isometry to model cones if the upper bound is attained.

The criterion is further refined by Pro in the context of a metric "sagitta" invariant, demonstrating that with additional control over critical points and distances, the maximal volume among all closed Riemannian $n$-manifolds with $sec \geq k,\, \operatorname{rad}\leq R$ and bounding sagitta $h$ is realized by lens spaces or their quotients (Pro, 2014).

2. Algebraic and Matrix Maximum Volume Substructures

In numerical linear algebra, the Maximum Volume Criterion is a central tool for robust low-rank matrix approximations, subset selection, and experimental design. Given $A \in \mathbb{R}^{m \times n}$ $(m \geq n)$, the (rectangular) volume is defined as

$$\mathrm{vol}(A) = \sqrt{\det(A^T A)} = \prod_{i=1}^n \sigma_i(A),$$

where $\sigma_i$ are the singular values. The search for a submatrix of maximal volume—particularly dominant or quasi-dominant submatrices under certain swap constraints—controls the stability and accuracy of pseudo-skeleton (CUR) decompositions (Mikhalev et al., 2015, Osinsky, 2018).

Algorithmic approaches such as maxvol, rect_maxvol, Dominant-C/R, and greedy RRQR variants achieve guaranteed proximity to the global volume maximum and permit spectral-norm error bounds and preconditioning guarantees that are near-optimal for fixed-size submatrix selection. These methods have direct connections to D-optimality in experimental design (i.e., maximizing $\det(A^T A)$ among all possible experiment row subsets), and have further implications for recommender systems and robust entrywise estimation in low-rank settings.

3. Complex and Hyperbolic Maximum Volume Models

In complex algebraic geometry, Berman and Berndtsson establish that among all toric Kähler–Einstein manifolds of dimension $n$, only the projective space $\mathbb{CP}^n$ attains the maximum volume (degree), bounded above by $(n+1)^n$. The proof leverages sharp Moser–Trudinger and Brezis–Merle inequalities for the complex Monge–Ampère operator and leads to a strict rigidity: equality holds only for $\mathbb{CP}^n$, with the bound sharp for Fano–index maximality (Berman et al., 2011).

For hyperbolic polyhedra, the supremum of possible volumes, given a fixed 1-skeleton (combinatorial type), is always achieved by the rectification of the graph, i.e., the right-angled ideal polyhedron whose skeleton comprises all combinatorial edges tangent to the boundary at infinity. Structural degenerations and volume-increasing deformations (as characterized by the Schl äfli formula) demonstrate that any alternative arrangement (with mixed vertex types or angle assignments) yields strictly lower volume (Belletti, 2020).

In representation theory of hyperbolic 3-manifolds, maximum volume rigidity theorems state that the only representations of the fundamental group attaining the hyperbolic volume are conjugate to the holonomy, both in dimension 3 (Mostow–Prasad rigidity) and in general $k$-manifolds for $m \geq k$ (Francaviglia et al., 2017).

4. Maximum Volume Criteria in Variational and Static Geometries

Critical metrics for the volume functional under scalar curvature and boundary constraints are sharply constrained by maximum volume bounds. For a compact $n$-manifold $(M,g)$ with smooth, connected, Einstein boundary and scalar curvature $n(n-1)\varepsilon$ ($\varepsilon \in \{-1,0,1\}$), geodesic balls in simply connected space forms attain the largest possible boundary volume for Miao–Tam critical metrics (Barros et al., 2017). Static metrics (vacuum solutions with positive cosmological constant) further exhibit that among all metrics with Einstein spherical boundary, the round hemisphere realizes maximal boundary volume—yielding a partial proof of the Cosmic no-hair conjecture in this category.

5. Maximum Volume Principles in Physical and Applied Contexts

Physical applications of the Maximum Volume Criterion are exemplified in wave energy absorption: the maximum power that line absorbers can extract from ocean waves under a volume constraint is characterized by two dimensionless parameters—device length and swept volume. The optimal absorbed power switches regimes depending on whether the volume constraint is active, and scales without limit in the long-device (infinite length) scenario. These results provide explicit guidelines for device design and explain observed super-linear power scaling effects (Stansell et al., 2011).

In droplet–fiber interactions, the maximum volume of a droplet that can stably adhere to a horizontal fiber is governed by a unified scaling law involving capillary length, fiber radius, and contact angle. The criterion provides a predictive envelope for engineering and natural phenomena involving capillary pinning, drop detachment, and morphological transitions (Zhang et al., 25 Dec 2025).

In safety verification and control, maximizing the volume of zonotopic invariant sets optimizes the certification region for system trajectories under affine dynamics. The true zonotope volume is a log-concave function of suitable parameterizations, allowing convex optimization methodologies to obtain the largest possible certified set subject to system and box constraints (Zhou et al., 21 May 2025).

6. Advanced Examples: Simplicial and Cosmological Extremal Volumes

The extremal partition of a simplex via cevians addresses the maximum possible sum of a specified subset of subsimplex volumes, expressed as a function of barycentric coordinates subject to unique root conditions on certain cubic polynomials. Explicit combinatorial and algebraic reduction yields sharp bounds and uniqueness for maximizing points, with closed-form solutions in low-dimensional examples (Guliyeva et al., 30 Dec 2025).

In cosmology, the maximum volume criterion identifies the largest scale for gravitational collapse in an inhomogeneous universe. By enforcing zero-averaged expansion within a chosen domain (as opposed to pointwise, as in the classical turnaround radius), one obtains analytical upper bounds for cluster size as a function of mass and redshift, grounded in Buchert’s scalar averaging formalism and the relativistic Zel’dovich approximation (Ostrowski et al., 2021).

7. Summary of Unifying Principles

Across these diverse domains, the Maximum Volume Criterion operates as a rigorous test: equality or near-equality with the model volume, or maximal attainability of the relevant volume functional, imposes strong structure—often rising to full geometric or algebraic rigidity. Table 1 summarizes exemplar models and their extremal behavior:

Context	Model Problem	Maximizer/Attainment
Sectional curvature/radius	$\sec \geq k$, $\mathrm{rad} \leq r$	Ball in $M_k^n$; rigidity if $|\operatorname{Vol}-V_k(r)| \to 0$
Alexandrov geometry	$f:Z \to X$ 1-Lipschitz, $\operatorname{vol}(Z)=\operatorname{vol}(X)$	$X = Z/\sim$ rigid quotient
Toric Kähler–Einstein geometry	Fano class, Kähler–Einstein metric	$\mathbb{CP}^n$, $\operatorname{Vol} = (n+1)^n$
Maximum matrix volume	Rectangular/square submatrix	Dominant/quasi-dominant, achieves volume within bound
Hyperbolic polyhedra	Fixed combinatorics (1-skeleton)	Rectification (right-angled ideal polyhedron)

The rigidity and extremality conferred by maximal volume conditions aligns with deep structural properties—extremal spaces are highly constrained, often unique, and serve as touchstones for further analysis and classification in geometry, topology, algebra, and applied mathematics.