Volume Power Functionals

• Volume power functionals are continuous, homogeneous maps that measure the 'size' or growth of geometric objects via nonlinear, log‐concave scaling laws.
• They capture asymptotic growth in algebraic geometry through divisor volumes and Okounkov bodies, showcasing properties like subadditivity and concavity.
• These functionals extend to convex, stochastic, and physical contexts, influencing designs in wave energy, porous media, and random geometric structures.

A volume power functional is a map defined on geometric or algebraic data—such as divisors, cycles, submanifolds, or simplicial complexes—that is continuous, homogeneous of prescribed degree $d > 0$, and typically log-concave, measuring “size” or “growth” via an (asymptotically) multiplicative, often nonlinear, scaling law on the underlying objects. The most prominent example in algebraic geometry is the volume function on (the numerical classes of) divisors, which records either the asymptotic rate of growth of sections or the Euclidean volume of an associated Okounkov body. In broader contexts, volume power functionals also capture, for example, generalizations on cycle classes, minimal energies in physics, geometric invariants in convex and metric geometry, and variance scaling of stochastic geometric structures.

1. Volume Power Functionals in Algebraic Geometry

For a smooth projective variety $X$ of dimension $n$ and a Cartier divisor $D$, the classical volume function

$$\operatorname{vol}_X(D) = \limsup_{k \to \infty} \frac{\dim H^0(X, \mathcal{O}_X(kD))}{k^n / n!}$$

encodes the asymptotic growth of sections and induces a map $$\operatorname{vol}_X: N^1(X)_\mathbb{R} \to \mathbb{R}_{\geq 0}$$. Three essential formal properties characterize this function:

• Continuity: The map extends to a continuous function on $N^1(X)_\mathbb{R}$.
• Degree $n$ Homogeneity: $$\operatorname{vol}_X(\lambda D) = \lambda^n \operatorname{vol}_X(D)$$ for all $\lambda \geq 0$.
• Log-Concavity: For big divisor classes $\xi, \xi'$,

$$\operatorname{vol}_X(\xi+\xi')^{1/n} \geq \operatorname{vol}_X(\xi)^{1/n} + \operatorname{vol}_X(\xi')^{1/n},$$

which reflects the Brunn–Minkowski property emerging from interpreting volumes as Euclidean volumes of convex Okounkov bodies.

The “volume power functional” $D \mapsto \operatorname{vol}(D)^{1/n}$ is thus subadditive and concave on the big cone.

Multigraded Setting: If $W_{(\bullet)}$ is a multigraded linear series indexed by $\mathbb{N}^\rho$ with $$m \mapsto W_m \subseteq H^0(X, \mathcal{O}_X(m_1 D_1 + \ldots + m_\rho D_\rho))$$, the generalized volume function

$$\operatorname{vol}_{W_{(\bullet)}}(a) = \limsup_{k \to \infty} \frac{\dim W_{ka}}{k^n / n!}$$

varies continuously, is homogeneous of degree $n$, and log-concave on the relevant cone.

Characterization:

• Any continuous, degree $n$ homogeneous, log-concave function on a closed convex cone $K \subseteq \mathbb{R}_+^\rho$ arises as the volume of some multigraded linear series on a toric variety.
• In contrast, the volume functions realized by complete linear series on projective varieties form only a countable set: the algebraic geometry of projective varieties imposes strong rigidity on the possible volume functions, whereas the multigraded setting can simulate arbitrary such functions, including “wild” ones (e.g., nowhere thrice differentiable).

2. Volume Power Functionals for Cycle Classes

The classical volume function on divisors generalizes in several ways to higher-codimension cycles (Lehmann, 2016):

• Mobility function: For a numerical $k$-cycle class $\alpha$, the mobility is defined as

$$\operatorname{mob}(\alpha) = \limsup_{m \to \infty} \frac{mc(m\alpha)}{m^{n/(n-k)}/n!},$$

where $mc(m\alpha)$ counts the maximum number of general points through which an effective family representing $m\alpha$ can pass.

• Intersection-theoretic volume:

$$\widehat{\operatorname{vol}}(\alpha) = \sup\{\operatorname{vol}(A) \;\mid\; \varphi_*[A^{n-k}] \preceq \alpha\},$$

where the supremum is over all birational models $\varphi:Y\to X$ and big and nef divisors $A$.

Both are volume power functionals: they are continuous, homogeneous of degree $n/(n-k)$, positive on exactly the big cone, and reflect enumerative or birational intersection-theoretic growth. For divisors, they coincide with the classical volume; in higher codimension, their equality is open.

3. Volume Power Functionals, Okounkov Bodies, and Transcendence

The log-concavity of volume power functionals is closely related to convex geometry via Okounkov bodies, with the formula

$$\operatorname{vol}_X(D) = n! \operatorname{vol}(\Delta_Y(D)).$$

Moreover, there are explicit examples in which the volume function (and thus its corresponding power functional) is given by a transcendental expression, as in the construction involving the projectivization of a bundle over $E\times E$ (with $E$ a general elliptic curve) and integrals over domains in $\mathbb{R}^2$ yielding transcendental values even for algebraically defined data (Kuronya et al., 2010).

4. Volume Power Functionals in Convex Geometry and Functional Spaces

Volume power functionals appear in convex geometry as functionals on convex bodies or on convex/coercive functions (Mussnig, 2018):

• Valuations on Convex Functions: For $u\in\mathrm{Conv}(\mathbb{R}^n)$,
  • Volume analogue: $$Z_{\mathrm{vol}}(u) = \int_{\mathrm{dom}\,u} \phi(u(x)) dx$$, continuous and homogeneous of degree $n$.
  • Polar volume analogue: $$Z_{\mathrm{pol}}(u) = \int_{\mathrm{dom}\,u^*} \phi_2(\nabla u^*(x)\cdot x - u^*(x))dx$$, homogeneous of degree $-n$.

These functionals extend the valuation theory and reflect the interplay between functional analysis and convex geometry, including invariance under $\mathrm{SL}(n)$ and translation, and degrees of homogeneity paralleling those of volume and polar volume.

5. Stochastic Geometry: Volume Power Functionals and Random Complexes

Volume power functionals serve as key statistics in the geometry of random structures:

• Intrinsic Volumes in Random Polytopes: Intrinsic volumes $\mathcal{L}_k$ (e.g., volume, surface area, Euler characteristic) of convex hulls or sublevel sets can be presented as integrals of polynomial functions of a defining function and its derivatives (Jubin, 2019). Their distributional properties, such as variance scaling (of order $s$ for parameter $s$), and rates of normal approximation are governed by stabilizing score representations and the Malliavin–Stein method, allowing for optimal Berry–Esseen bounds (Lachièze-Rey et al., 2017).
• Random Simplicial Complexes: In the context of random Vietoris–Rips complexes, the volume power functional

$$V_k^{(\alpha)} = \frac{1}{(k+1)!} \sum \left[\Delta_{\delta_t}(x_0, ..., x_k)\right]^{\alpha}$$

collects $\alpha$-th powers of simplex volumes. The asymptotic covariance structure of vectors of such functionals reveals a sharp dichotomy between regimes: - Supercritical: Covariance is rank one (degenerate), dominated by a single direction in function space. - Subcritical: Covariance is full rank and positive definite. - Critical: Covariance is sums of weighted matrices, with detailed eigenstructure (Westenholz, 19 Sep 2025).

This spectral characterization has direct stochastic implications, including variance bounds, regression structure, and central limit theorems.

6. Applications in Physics and Engineering

Volume power functionals are integral to the modeling of physical or engineering systems:

• Wave Energy Conversion: Absorbed power and efficiency of wave energy converters are governed by nonlinear relations (volume power functionals) of the available swept volume and device geometry, with scaling laws manifesting as upper bounds (Budal’s bound) and design guidelines for optimal performance (Stansell et al., 2011).
• Permeability in Porous Media: The permeability $k$ in 3D porous systems obeys power-law scaling with four Minkowski functionals: volume (porosity), critical cross-sectional area, integral mean curvature, and Euler characteristic. A combinatoric function $F$ weaving these invariants yields a universal scaling

$k = C \cdot F^{0.428},$

independent of particle size/distribution and characterizing both ordered and disordered structures (Haque et al., 8 Oct 2024).

• Reduced Density-Matrix Functional Theory: In electronic structure, power functionals $f(n, n′) = (nn′)^\alpha$ interpolate between mean-field and correlated regimes. Their physical validity is controlled by parametric constraints (Pauli principle, Lieb–Oxford bound), only holding in specific density/parameter regions, illustrating the fine balance between functional simplicity and physical reality (Putaja et al., 2016).
• Gauge and Holonomy Theories: In the mirror-symmetric theory, the Dirac–Born–Infeld action (a volume power functional on Hermitian connections) is minimized by flat or calibrated connections, with topological quantization implications in manifolds of special holonomy (Kawai et al., 2021).

7. Regularity and Analytic Properties

The volume function and associated power functionals possess subtle regularity properties. The volume function on the big cone of a projective manifold is $C^{1,1}$, i.e., differentiable with locally Lipschitz gradient, but may fail to be $C^{2,\alpha}$ even along segments moving in ample directions. This optimal regularity has concrete implications: it controls the smoothness of the volume power functional $$t \mapsto \operatorname{vol}(\alpha + t\omega)^{1/n}$$ and thus determines the tractability of related asymptotics and variational problems (Cao et al., 20 May 2025).

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In total, volume power functionals thread through diverse mathematical and applied fields as nonlinear, scale-sensitive, often concave functionals encoding the essential “size” or “growth” properties of algebraic, geometric, stochastic, and even analytic objects. While in algebraic geometry their structural properties and regularity rigidly constrain their possible forms (reflecting deep properties of varieties), in stochastic or physical settings they determine scaling laws, central limit behavior, and variational optima, unified by the paradigm of homogeneous, often log-concave, functionals homogenizing additive and multiplicative geometric invarants.