Dual Volume Representations

• Dual volume representations are constructions where volumes are encoded on dual objects, effectively transforming and regularizing analytic and geometric measures.
• They have diverse applications across holography, convex and hyperbolic geometry, positive geometries, and computational algorithms to reveal invariants and variational duality.
• The dual approach converts challenging primal volume problems into more tractable dual formulations, yielding novel interpretations and computational advances.

Dual volume representations are constructions in which a geometric, physical, or informational quantity is encoded by a volume attached to a dual object, a regularized difference of volumes, or a measure supported on a dual domain. The expression is not tied to a single theory. In current arXiv usage it spans holography, where bulk volumes are related to fidelity susceptibility and thermodynamic volume; convex and discrete geometry, where polarity converts primal volume problems into dual ones; hyperbolic geometry, where convex-core volume is corrected by boundary bending to produce a dual volume; and positive-geometry theory, where canonical functions are represented as Laplace transforms on convex dual cones (Momeni et al., 2017, Schlenker, 2019, Gao et al., 2024, Mazzucchelli et al., 2 Sep 2025).

1. Conceptual scope and recurrent mechanisms

Across these literatures, the common structural feature is a passage from a “primal” object to a dual datum on which volume is simpler, more canonical, or more tightly connected to invariants of interest. In convexity, the dual object is usually a polar body or polytope, and set inclusion reverses under polarity. In holography, different bulk volumes are assigned different information-theoretic duals. In hyperbolic 3-manifolds, a dual volume subtracts a boundary term from the ordinary convex-core volume. In positive geometries, a canonical rational function is represented by a measure on the convex dual cone (Abdolali et al., 2024, Momeni et al., 2017, Schlenker, 2019, Mazzucchelli et al., 2 Sep 2025).

Domain	Primal object	Dual volume construction
Holography	deformed AdS bulk region	regularized maximal/minimal bulk volume
Convex geometry	simplex, polytope, star body	polar body, dual mixed volume, dual polytope
Hyperbolic 3-manifolds	convex core	$V_C^* = V_C - \frac{1}{2}L_m(l)$
Positive geometries	canonical form/function	Laplace transform on the dual cone
SSMF	minimum-volume enclosing simplex	maximum-volume dual simplex

A recurring implication is that dualization changes not only representation but also the analytic character of the problem. Minimum-volume optimization can become maximum-volume optimization in the dual space; an infinite or divergent volume can become finite after subtraction of a canonical background term; and rational canonical functions can become moment-generating objects for measures on a dual cone. This suggests that “dual volume representation” is less a single definition than a family of methods organized around polarity, regularization, and variational duality.

2. Holographic information, complexity, and thermodynamic volume

In the AdS black-hole setting, thermodynamic and information-theoretic duals of volume were analyzed by regularizing divergent bulk volumes through background subtraction (Momeni et al., 2017). For an AdS-Schwarzschild black hole in extended thermodynamic phase space, the thermodynamic volume is

$V=\frac{4}{3}\pi r_+^3,$

and the pressure is

$P=\frac{3}{8\pi l^2}.$

The same work defines holographic complexity by

$\mathcal{C}=\frac{V(\gamma)}{8\pi RG},$

with $V(\gamma)$ the volume inside the minimal surface $\gamma$, and regularizes it by

$$\Delta \mathcal{C}=\frac{V(\gamma)_{DAdS}-V(\gamma)_{AdS}}{8\pi RG}.$$

The regularized fidelity susceptibility is similarly defined from the maximal bulk time slice,

$$\Delta \chi_F=\frac{V(\Sigma_{\max})_{DAdS}-V(\Sigma_{\max})_{AdS}}{8\pi RG}.$$

The central claim of the 2017 analysis is that, for all black-hole backgrounds studied—Schwarzschild-AdS, Reissner–Nordström AdS, and higher-dimensional Schwarzschild-AdS—the regularized fidelity susceptibility exhibits the same behavior as the thermodynamic volume, while the regularized holographic complexity behaves very differently (Momeni et al., 2017). The paper states that both thermodynamic volume and regularized fidelity susceptibility give rise to similar, essentially identical up to normalization, $PV$ equations of state, whereas the $P_{com}$–$V=\frac{4}{3}\pi r_+^3,$0 relation from holographic complexity is qualitatively different. On that basis, it argues that the thermodynamic volume is best understood as holographically dual to the regularized maximal volume, namely fidelity susceptibility.

That proposal is not uncontroversial. A subsequent analysis of the conjecture that fidelity susceptibility is exactly dual to maximal-slice volume found that the leading and subleading divergences of the two quantities have similar structure but inconsistent numerical coefficients in the examples studied (Moosa et al., 2018). For $V=\frac{4}{3}\pi r_+^3,$1, both the CFT fidelity susceptibility and the maximal AdS volume contain $V=\frac{4}{3}\pi r_+^3,$2, $V=\frac{4}{3}\pi r_+^3,$3, and $V=\frac{4}{3}\pi r_+^3,$4 divergences, but a single cutoff-independent proportionality constant does not reconcile all coefficients simultaneously. The later paper therefore supports similarity of divergence structure while arguing against an exact universal identification in the strong sense. A common misconception is therefore that maximal bulk volume and fidelity susceptibility have been established as exactly equal; the cited analyses support a weaker picture in which regularized bulk-volume behavior tracks thermodynamic or information-theoretic data, but exact equality remains unsettled (Momeni et al., 2017, Moosa et al., 2018).

3. Dual Brunn–Minkowski theory and polar-volume constructions

In convex geometry, dual volume representations are foundational rather than derivative. The dual Brunn–Minkowski theory replaces Minkowski addition by radial addition on star sets and defines the dual mixed volume by

$V=\frac{4}{3}\pi r_+^3,$5

where $V=\frac{4}{3}\pi r_+^3,$6 is the radial function (Dulio et al., 2013). Dulio, Gardner, and Peri show that this object is characterized by additivity, rotation invariance, and vanishing whenever the intersection of two arguments is $V=\frac{4}{3}\pi r_+^3,$7; under those assumptions a functional must be a constant multiple of the dual mixed volume. They also prove that the characterization is best possible in the sense that none of the assumptions can be omitted (Dulio et al., 2013).

The same dual framework admits Orlicz generalization. For star bodies $V=\frac{4}{3}\pi r_+^3,$8, the Orlicz $V=\frac{4}{3}\pi r_+^3,$9-dual mixed volume is

$P=\frac{3}{8\pi l^2}.$0

and is tied to an Orlicz $P=\frac{3}{8\pi l^2}.$1-radial addition that extends classical radial and $P=\frac{3}{8\pi l^2}.$2-radial addition (Ye, 2014). The resulting theory includes dual Orlicz–Brunn–Minkowski, dual Orlicz–Minkowski, dual Orlicz isoperimetric, and dual Orlicz–Urysohn inequalities. Here, the dual volume is no longer merely a polar-volume correction; it is an Orlicz-weighted radial integral over the sphere (Ye, 2014).

A different polar construction appears in the 2024 definition of the dual mixed volume rational function. For polytopes $P=\frac{3}{8\pi l^2}.$3,

$P=\frac{3}{8\pi l^2}.$4

which is a homogeneous rational function of degree $P=\frac{3}{8\pi l^2}.$5, in contrast with the classical mixed-volume polynomial of degree $P=\frac{3}{8\pi l^2}.$6 (Gao et al., 2024). This dual mixed volume is additive under mixed subdivisions and is related, after a change of variables, to the dual volume of the Cayley polytope. The same paper computes explicit formulas for zonotopes, generalized permutohedra, and associahedra, and states that the associahedral formula reproduces the planar $P=\frac{3}{8\pi l^2}.$7-scalar amplitude at tree level (Gao et al., 2024).

Duality also yields extremal volume bounds. For a $P=\frac{3}{8\pi l^2}.$8-dimensional canonical Fano polytope $P=\frac{3}{8\pi l^2}.$9 with exactly one interior lattice point and $\mathcal{C}=\frac{V(\gamma)}{8\pi RG},$0,

$\mathcal{C}=\frac{V(\gamma)}{8\pi RG},$1

where $\mathcal{C}=\frac{V(\gamma)}{8\pi RG},$2 is the $\mathcal{C}=\frac{V(\gamma)}{8\pi RG},$3-st Sylvester number, and equality holds if and only if $\mathcal{C}=\frac{V(\gamma)}{8\pi RG},$4 is unimodularly equivalent to a specific reflexive simplex $\mathcal{C}=\frac{V(\gamma)}{8\pi RG},$5 (Balletti et al., 2016). In toric terms this gives a sharp upper bound on the anti-canonical degree $\mathcal{C}=\frac{V(\gamma)}{8\pi RG},$6 for toric Fano varieties with at worst canonical singularities. Here the dual volume is itself the principal extremal invariant (Balletti et al., 2016).

4. Polytopes, canonical forms, and measure-theoretic duality

For specific polytope families, dual volume representations are often completely combinatorial. In the case of twinned chain polytopes built from two posets $\mathcal{C}=\frac{V(\gamma)}{8\pi RG},$7 and $\mathcal{C}=\frac{V(\gamma)}{8\pi RG},$8, the volume of the primal polytope is

$\mathcal{C}=\frac{V(\gamma)}{8\pi RG},$9

and each facet-supporting hyperplane has the form

$V(\gamma)$0

for a maximal chain $V(\gamma)$1 in some ordinal sum $V(\gamma)$2 (Tsuchiya, 2015). The vertices of the dual polytope are then explicitly the primitive integer normals coming from those facets: $V(\gamma)$3 The geometry of the dual is thus determined by maximal chains across all induced ordinal sums of the two posets (Tsuchiya, 2015).

Positive-geometry theory recasts such constructions in a more analytic language. The 2025 paper “Canonical Forms as Dual Volumes” studies positive geometries in projective spaces whose canonical functions can be expressed as Laplace transforms of measures supported on the convex dual of the associated semialgebraic set: $V(\gamma)$4 When $V(\gamma)$5 is non-negative, the geometry is called completely monotone, reflecting complete monotonicity of the canonical function (Mazzucchelli et al., 2 Sep 2025). The paper states that completely monotone positive geometries are characterized by algebraic boundaries cut out by hyperbolic polynomials, with the geometry equal to a hyperbolicity region, and that simplex-like minimal spectrahedra are completely monotone, with representing measures related to the Wishart distribution (Mazzucchelli et al., 2 Sep 2025).

This measure-theoretic dualization places canonical forms, hyperbolicity, and convex duality in a common framework. A polytope case already fits this pattern: $V(\gamma)$6 The 2025 work extends that picture to geometries bounded by lines and conics and to a nodal cubic, where the representing measures can involve transcendental periods rather than purely rational densities (Mazzucchelli et al., 2 Sep 2025). A plausible implication is that dual volume representations interpolate between discrete combinatorics, convex algebraic geometry, and the theory of positive geometries rather than belonging exclusively to any one of them.

5. Hyperbolic 3-manifolds, renormalized volume, and rigidity-type duality

In quasifuchsian hyperbolic 3-manifolds, the convex core has infinite-volume ambient context but a finite intrinsic correction called the dual volume: $V(\gamma)$7 where $V(\gamma)$8 is the length of the measured bending lamination on $V(\gamma)$9 with respect to the induced metric $\gamma$0 (Schlenker, 2019). For a smooth convex subset $\gamma$1, the analogous formula is

$\gamma$2

This is the direct geometric analogue of subtracting a boundary term to produce a more natural variational object (Schlenker, 2019).

The principal variational formula is the dual Bonahon–Schläfli identity,

$\gamma$3

which mirrors the variational formula for the renormalized volume $\gamma$4 at infinity (Schlenker, 2019). The same paper emphasizes two simultaneous facts: the dual volume has striking similarities with the renormalized volume, especially in their variational formulas, and the two remain within bounded additive distance. More precisely,

$\gamma$5

It also records upper bounds in terms of the Weil–Petersson distance between the conformal metrics at infinity (Schlenker, 2019).

A related but distinct dual perspective on volume appears in bounded-cohomological rigidity theory. Bucher, Burger, and Iozzi define the volume of an $\gamma$6-valued representation of a hyperbolic lattice by pairing the pullback of a bounded volume class with a relative fundamental class,

$\gamma$7

and present this as a dual counterpart to the simplicial-volume proof of Mostow rigidity (Bucher et al., 2012). This is not a dual volume in the polar or convex-core sense, but it belongs to the same family of ideas in which geometric volume is reconstructed from a dual object—in this case bounded cohomology rather than a dual body or dual cone (Bucher et al., 2012).

6. Algorithmic and computational formulations

In optimization and data analysis, dual volume representations often convert an intractable primal enclosure problem into a tractable dual one. For simplex-structured matrix factorization, the 2024 dual formulation uses polarity of polytopes to convert minimum-volume SSMF into a maximum-volume problem in dual space (Abdolali et al., 2024). After translating the data so that the origin lies in the convex hull and projecting to the intrinsic affine span, the dual constraint becomes

$\gamma$8

and the optimization problem is

$\gamma$9

The paper proves identifiability of this maximum-volume dual problem under the same sufficiently scattered condition used for minimum-volume SSMF, and uses block coordinate ascent plus a min-max translation update to bridge volume-minimization and facet-identification methods (Abdolali et al., 2024).

A second algorithmic instance is dual volume sampling. For a short-and-wide matrix $$\Delta \mathcal{C}=\frac{V(\gamma)_{DAdS}-V(\gamma)_{AdS}}{8\pi RG}.$$0 and $$\Delta \mathcal{C}=\frac{V(\gamma)_{DAdS}-V(\gamma)_{AdS}}{8\pi RG}.$$1, a $$\Delta \mathcal{C}=\frac{V(\gamma)_{DAdS}-V(\gamma)_{AdS}}{8\pi RG}.$$2-subset $$\Delta \mathcal{C}=\frac{V(\gamma)_{DAdS}-V(\gamma)_{AdS}}{8\pi RG}.$$3 is sampled with probability

$$\Delta \mathcal{C}=\frac{V(\gamma)_{DAdS}-V(\gamma)_{AdS}}{8\pi RG}.$$4

The normalization constant is

$$\Delta \mathcal{C}=\frac{V(\gamma)_{DAdS}-V(\gamma)_{AdS}}{8\pi RG}.$$5

and the paper gives an exact randomized polynomial-time sampling algorithm with complexity $$\Delta \mathcal{C}=\frac{V(\gamma)_{DAdS}-V(\gamma)_{AdS}}{8\pi RG}.$$6, a derandomization, and a proof that the induced measure is Strongly Rayleigh via real stable polynomials (Li et al., 2017). One consequence is a fast-mixing Markov chain with

$$\Delta \mathcal{C}=\frac{V(\gamma)_{DAdS}-V(\gamma)_{AdS}}{8\pi RG}.$$7

Here the “dual volume” is the determinant of the row-span Gram matrix of the chosen submatrix, and the dual formulation is algorithmically central rather than merely interpretive (Li et al., 2017).

A broader computational usage concerns dual volumetric structures rather than dual volumes in the strict polar sense. Dual Octree Graph Networks encode a 3D shape by an adaptive feature volume organized by an octree and process graph convolutions on the dual graph of face-adjacent octree leaves, including cross-scale adjacency (Wang et al., 2022). Generalised primal-dual grids for co-volume schemes extend Delaunay–Voronoi structures to weighted Regular-Power tessellations optimized over positions, topology, and scalar weights (Engwirda, 2017). The 2025 FVE-2L method introduces a two-layer dual strategy with a fixed number of four dual elements per triangle, $$\Delta \mathcal{C}=\frac{V(\gamma)_{DAdS}-V(\gamma)_{AdS}}{8\pi RG}.$$8 regularity for the $$\Delta \mathcal{C}=\frac{V(\gamma)_{DAdS}-V(\gamma)_{AdS}}{8\pi RG}.$$9 theory, and local conservation on all second-layer dual elements (Huang et al., 3 Mar 2025). These examples use “dual” and “volume” differently from convex duality or holography, but they show that the language of dual volumetric representation has become technically productive in numerical geometry, PDE discretization, and 3D learning as well.

In that expanded sense, dual volume representations occupy a spectrum. At one end lie exact geometric dualities—polar bodies, dual mixed volumes, convex-core dual volume, and Laplace-transform measures on dual cones. At the other lie computational architectures and discretizations built from dual simplices, dual graphs, or dual control volumes. The literature collectively indicates that volume becomes most informative when paired with a dual object whose structure exposes invariants, regularizes divergences, or converts a primal problem into a better-posed dual one (Gao et al., 2024, Mazzucchelli et al., 2 Sep 2025, Abdolali et al., 2024).