Causal Convexity: A Multi-Disciplinary Perspective

• Causal convexity is the study of how imposing causality constraints on systems leads to diverse convex structures across frameworks like relativistic hydrodynamics, order theory, and optimization.
• In relativistic hydrodynamics, convexity is governed by the relativistic fundamental derivative, ensuring genuine nonlinearity and stable wave propagation.
• The concept underpins polyhedral causal correlations, convex relaxations in causal inference, and duality in adapted transport, enabling efficient optimization and robust analysis.

Searching arXiv for recent and relevant papers on causal convexity across the distinct technical senses represented in the provided corpus. Causal convexity denotes a family of technically distinct convexity notions that arise when causality constraints are imposed on dynamical systems, ordered spaces, correlation sets, partial-identification programs, and transport couplings. In relativistic hydrodynamics it refers to Lax convexity of the hyperbolic perfect-fluid system under a causal equation of state, governed by a relativistically corrected fundamental derivative (Ibáñez et al., 2013). In topological preorder theory and spacetime causality it refers to order-convex subsets and neighborhoods, with consequences for quasi-uniformizability and quasi-pseudo-metrizability (Minguzzi, 2012). In multipartite correlation theory, partial identification, and adapted transport it appears as convex polytopes, convex relaxations, convex Pareto frontiers, and convex sets of causal, bicausal, or multicausal couplings (Abbott et al., 2016, Zeitler et al., 2022, Javurek et al., 11 May 2026, Kršek et al., 2024).

1. Relativistic hydrodynamic convexity

In general relativistic hydrodynamics for a perfect fluid with equation of state $p=p(\rho,\varepsilon)$, the governing equations are the conservation laws

$$\nabla_\mu J^\mu = 0, \qquad \nabla_\mu T^{\mu\nu}=0,$$

with

$$J^\mu=\rho u^\mu, \qquad T^{\mu\nu}=\rho h\,u^\mu u^\nu + p\, g^{\mu\nu},$$

and

$h=1+\varepsilon+\frac{p}{\rho}.$

For a causal equation of state, meaning $c_s^2<1$, the relativistic hydrodynamic system is hyperbolic. The convexity question is then posed in the sense of nonlinear hyperbolic conservation laws: if $\lambda_\alpha$ are characteristic speeds and $\mathbf r_\alpha$ the corresponding right eigenvectors, the relevant Lax indicator is

$$\mathcal P_\alpha=\nabla_{\mathbf u}\lambda_\alpha\cdot \mathbf r_\alpha.$$

A field is genuinely nonlinear when $\mathcal P_\alpha\neq 0$, linearly degenerate when $\mathcal P_\alpha=0$, and the system is convex when all characteristic fields are of one of these two types without sign changes in $$\nabla_\mu J^\mu = 0, \qquad \nabla_\mu T^{\mu\nu}=0,$$0 nearby (Ibáñez et al., 2013).

For the acoustic fields of GRHD, the central thermodynamic factor reduces to

$$\nabla_\mu J^\mu = 0, \qquad \nabla_\mu T^{\mu\nu}=0,$$1

where $$\nabla_\mu J^\mu = 0, \qquad \nabla_\mu T^{\mu\nu}=0,$$2 is the classical fundamental derivative. Equivalently, one defines the relativistic fundamental derivative

$$\nabla_\mu J^\mu = 0, \qquad \nabla_\mu T^{\mu\nu}=0,$$3

The main criterion is

$$\nabla_\mu J^\mu = 0, \qquad \nabla_\mu T^{\mu\nu}=0,$$4

This is stricter than the classical gas-dynamical condition $$\nabla_\mu J^\mu = 0, \qquad \nabla_\mu T^{\mu\nu}=0,$$5. In the nonrelativistic limit, $$\nabla_\mu J^\mu = 0, \qquad \nabla_\mu T^{\mu\nu}=0,$$6 and $$\nabla_\mu J^\mu = 0, \qquad \nabla_\mu T^{\mu\nu}=0,$$7, so $$\nabla_\mu J^\mu = 0, \qquad \nabla_\mu T^{\mu\nu}=0,$$8, recovering the Menikoff–Plohr criterion.

Loss of convexity has direct wave-dynamical meaning. The paper ties non-convex states to anomalous behavior in which rarefaction waves need not remain expansive; they may become compressive, the acoustic fields may lose genuine nonlinearity, and nonclassical wave phenomena may appear. In one-dimensional self-similar isentropic special relativistic flow,

$$\nabla_\mu J^\mu = 0, \qquad \nabla_\mu T^{\mu\nu}=0,$$9

and

$$J^\mu=\rho u^\mu, \qquad T^{\mu\nu}=\rho h\,u^\mu u^\nu + p\, g^{\mu\nu},$$0

so the sign of $$J^\mu=\rho u^\mu, \qquad T^{\mu\nu}=\rho h\,u^\mu u^\nu + p\, g^{\mu\nu},$$1 controls whether the acoustic families retain the expansive character associated with convexity.

2. Order-topological and spacetime causal convexity

In the theory of closed preordered spaces, the basic object is a topological preordered space $$J^\mu=\rho u^\mu, \qquad T^{\mu\nu}=\rho h\,u^\mu u^\nu + p\, g^{\mu\nu},$$2, with increasing and decreasing hulls

$$J^\mu=\rho u^\mu, \qquad T^{\mu\nu}=\rho h\,u^\mu u^\nu + p\, g^{\mu\nu},$$3

and, for a subset $$J^\mu=\rho u^\mu, \qquad T^{\mu\nu}=\rho h\,u^\mu u^\nu + p\, g^{\mu\nu},$$4,

$$J^\mu=\rho u^\mu, \qquad T^{\mu\nu}=\rho h\,u^\mu u^\nu + p\, g^{\mu\nu},$$5

A subset $$J^\mu=\rho u^\mu, \qquad T^{\mu\nu}=\rho h\,u^\mu u^\nu + p\, g^{\mu\nu},$$6 is convex when

$$J^\mu=\rho u^\mu, \qquad T^{\mu\nu}=\rho h\,u^\mu u^\nu + p\, g^{\mu\nu},$$7

and a closed preorder means that the graph

$$J^\mu=\rho u^\mu, \qquad T^{\mu\nu}=\rho h\,u^\mu u^\nu + p\, g^{\mu\nu},$$8

is closed in $$J^\mu=\rho u^\mu, \qquad T^{\mu\nu}=\rho h\,u^\mu u^\nu + p\, g^{\mu\nu},$$9. Convexity at a point $h=1+\varepsilon+\frac{p}{\rho}.$0 requires that every open neighborhood $h=1+\varepsilon+\frac{p}{\rho}.$1 contain a set of the form $h=1+\varepsilon+\frac{p}{\rho}.$2, where $h=1+\varepsilon+\frac{p}{\rho}.$3 is open decreasing and $h=1+\varepsilon+\frac{p}{\rho}.$4 is open increasing. Local convexity means that convex neighborhoods form a neighborhood base (Minguzzi, 2012).

The central structural result is that, for locally compact $h=1+\varepsilon+\frac{p}{\rho}.$5-compact $h=1+\varepsilon+\frac{p}{\rho}.$6-preordered spaces, local convexity promotes to global convexity: every locally convex $h=1+\varepsilon+\frac{p}{\rho}.$7-preordered locally compact $h=1+\varepsilon+\frac{p}{\rho}.$8-compact space is a convex normally preordered space, and hence quasi-uniformizable. The paper also identifies broad hypotheses under which local convexity holds. One is the $h=1+\varepsilon+\frac{p}{\rho}.$9-preserving condition, namely that every compact set $c_s^2<1$0 has compact convex hull

$c_s^2<1$1

Another is compact generation of the preorder: there exists $c_s^2<1$2 such that $c_s^2<1$3 is compact for every compact $c_s^2<1$4, and $c_s^2<1$5 is the smallest closed preorder containing $c_s^2<1$6. Under antisymmetry and local compactness, these hypotheses yield weak convexity and then convexity.

In spacetime applications, order-convexity becomes causal convexity. For a causal preorder such as $c_s^2<1$7, $c_s^2<1$8, or $c_s^2<1$9, a set $\lambda_\alpha$0 is convex exactly when every intermediate point of a causal chain between two points of $\lambda_\alpha$1 remains in $\lambda_\alpha$2. The paper shows that every stably causal spacetime is quasi-uniformizable and every globally hyperbolic spacetime is strictly quasi-pseudo-metrizable. It also emphasizes a negative result: local compactness, $\lambda_\alpha$3-compactness, and closedness of the order do not by themselves imply convexity. The counterexample on $\lambda_\alpha$4 shows that closed order plus tame topology is insufficient to force local causal convexity.

3. Causal polytopes in multipartite correlation theory

In multipartite nonlocality and indefinite-causal-order theory, causal convexity refers to the structure of the set of correlations compatible with definite, though possibly random and dynamical, causal order. In a fixed finite Bell-type scenario with parties $\lambda_\alpha$5, finite inputs $\lambda_\alpha$6, and finite outputs $\lambda_\alpha$7, a correlation is a conditional distribution $\lambda_\alpha$8. The paper defines $\lambda_\alpha$9-partite causal correlations recursively: for $\mathbf r_\alpha$0,

$\mathbf r_\alpha$1

where $\mathbf r_\alpha$2, $\mathbf r_\alpha$3, $\mathbf r_\alpha$4 is a single-party distribution, and each conditioned remainder $\mathbf r_\alpha$5 is a causal $\mathbf r_\alpha$6-partite correlation. This recursive form allows later causal order to depend on earlier inputs and outputs, so it includes genuinely dynamical order (Abbott et al., 2016).

The set of such correlations is convex: if $\mathbf r_\alpha$7 and $\mathbf r_\alpha$8 are causal, then $\mathbf r_\alpha$9 is causal. More strongly, in every fixed finite scenario the set is a convex polytope. Every causal correlation can be decomposed as a convex combination of deterministic causal correlations,

$$\mathcal P_\alpha=\nabla_{\mathbf u}\lambda_\alpha\cdot \mathbf r_\alpha.$$0

with finitely many deterministic strategies because the alphabets are finite. The vertices correspond to deterministic strategies, and the nontrivial facets define causal inequalities.

The simplest nontrivial tripartite example illustrates the geometry. In the scenario with three parties, binary inputs, and the asymmetric binary-output convention described in the paper, the causal polytope has $$\mathcal P_\alpha=\nabla_{\mathbf u}\lambda_\alpha\cdot \mathbf r_\alpha.$$1 vertices, of which $$\mathcal P_\alpha=\nabla_{\mathbf u}\lambda_\alpha\cdot \mathbf r_\alpha.$$2 are compatible with fixed causal order and $$\mathcal P_\alpha=\nabla_{\mathbf u}\lambda_\alpha\cdot \mathbf r_\alpha.$$3 require dynamical order; it is $$\mathcal P_\alpha=\nabla_{\mathbf u}\lambda_\alpha\cdot \mathbf r_\alpha.$$4-dimensional and has $$\mathcal P_\alpha=\nabla_{\mathbf u}\lambda_\alpha\cdot \mathbf r_\alpha.$$5 facets grouped into $$\mathcal P_\alpha=\nabla_{\mathbf u}\lambda_\alpha\cdot \mathbf r_\alpha.$$6 equivalence classes. This establishes that causality in this sense is broader than convex mixtures of fixed global orders. The same paper also shows that some tripartite causal inequalities are violated in the process matrix formalism, where quantum mechanics is locally valid but no global causal structure is assumed.

4. Convex relaxations and convex frontiers in causal inference

In discrete partial identification, the “causal marginal polytope” is a convex relaxation of exact global SCM optimization. Instead of searching over a single globally realizable response-function model, the method introduces a collection of smaller marginal causal models $$\mathcal P_\alpha=\nabla_{\mathbf u}\lambda_\alpha\cdot \mathbf r_\alpha.$$7 on subsets of variables, each with local latent parameters constrained to simplices, required to reproduce the available observational or interventional distributions, and required to agree on overlaps. Because local implied probabilities are linear in the local parameters, simplex constraints are linear, data-binding constraints are linear, overlap-consistency equalities are linear, and the optional weak-edge and weak-confounding assumptions are linear inequalities, the feasible region is a convex polytope. Treatment-effect functionals such as

$$\mathcal P_\alpha=\nabla_{\mathbf u}\lambda_\alpha\cdot \mathbf r_\alpha.$$8

remain linear over this region, so lower and upper bounds are obtained by linear programming. The construction is explicitly a relaxation: local consistency does not in general imply existence of a single global SCM, and the resulting bounds are therefore generally outer bounds rather than exact Balke–Pearl bounds (Zeitler et al., 2022).

A distinct use of convexity appears in causal sensitivity analysis under hidden confounding. There the bounds

$$\mathcal P_\alpha=\nabla_{\mathbf u}\lambda_\alpha\cdot \mathbf r_\alpha.$$9

are treated as forward optimization problems over the set of full distributions $\mathcal P_\alpha\neq 0$0 that are observationally compatible and satisfy a generalized treatment sensitivity model constraint

$\mathcal P_\alpha\neq 0$1

The underlying bi-objective program is

$\mathcal P_\alpha\neq 0$2

Under the standard convexity and linearity conditions that $\mathcal P_\alpha\neq 0$3 is convex in $\mathcal P_\alpha\neq 0$4 and $\mathcal P_\alpha\neq 0$5 is linear in $\mathcal P_\alpha\neq 0$6, the Lagrangian scalarization

$\mathcal P_\alpha\neq 0$7

recovers the full Pareto frontier of solutions. The paper uses this convex geometry to generate labels for prior-data fitted networks and reports test-time computation that is orders of magnitude faster than per-instance methods (Javurek et al., 11 May 2026).

These two strands treat convexity differently. In the causal marginal polytope, convexity is a polyhedral relaxation over locally consistent causal marginals. In amortized sensitivity analysis, convexity is the geometry of an ambiguity set and of a Pareto frontier over observationally compatible latent distributions. A plausible implication is that “causal convexity” in causal inference is less a single doctrine than a family of optimization geometries used to make partial identification computationally tractable.

5. Causal, bicausal, and multicausal transport

In adapted optimal transport, causality is imposed directly on couplings. In the path-space formulation, a coupling $\mathcal P_\alpha\neq 0$8 between $\mathcal P_\alpha\neq 0$9 and $\mathcal P_\alpha=0$0 is causal if

$\mathcal P_\alpha=0$1

for all $\mathcal P_\alpha=0$2. Bicausality is causality in both directions. In the multimarginal setting, a coupling is multicausal if each process $\mathcal P_\alpha=0$3 is conditionally independent of the other processes’ information up to time $\mathcal P_\alpha=0$4, given its own past up to $\mathcal P_\alpha=0$5. The corresponding primal problems minimize a linear cost functional over the admissible couplings, for example

$\mathcal P_\alpha=0$6

with analogous definitions for bicausal and multicausal transport (Kršek et al., 2024).

The paper places these problems in a general convex-dual framework. Causal, bicausal, and multicausal admissibility are encoded by linear test-function equalities against adapted martingale-difference classes $\mathcal P_\alpha=0$7, $\mathcal P_\alpha=0$8, and $\mathcal P_\alpha=0$9. The duals therefore take the form

$$\nabla_\mu J^\mu = 0, \qquad \nabla_\mu T^{\mu\nu}=0,$$00

and analogously in the bicausal and multimarginal cases. Under minimal assumptions—measurability of the cost and an integrable upper bound—the paper proves strong duality and dual attainment for causal and bicausal transport, for multimarginal causality-constrained transport, and for causal and bicausal barycenter problems.

A further geometric result is the characterization of polar sets: subsets that are assigned zero mass by every admissible causal, bicausal, or multicausal coupling. These sets are described by stagewise full-measure rectangular exclusions adapted to the relevant filtrations. In robust finance, the same structure acquires a market interpretation. A no-arbitrage condition in a non-dominated model of several financial markets naturally leads to multicausal couplings, and the robust superhedging price becomes

$$\nabla_\mu J^\mu = 0, \qquad \nabla_\mu T^{\mu\nu}=0,$$01

Here causal convexity is not merely set-theoretic; it is the dualizable geometry of admissible couplings under nonanticipativity.

6. Comparative structure, misconceptions, and scope

Across these literatures, causal convexity does not denote a single invariant mathematical object. In relativistic hydrodynamics it is convexity of a hyperbolic PDE system controlled by $$\nabla_\mu J^\mu = 0, \qquad \nabla_\mu T^{\mu\nu}=0,$$02; in preorder spaces it is order-convexity and the globalization of local convexity; in multipartite correlation theory it is the polyhedral structure of definite-causal-order correlations; in causal inference it is either a polyhedral relaxation or a convex Pareto frontier over observationally compatible latent distributions; and in adapted transport it is the convex-dual structure of causality-constrained couplings (Ibáñez et al., 2013, Minguzzi, 2012, Abbott et al., 2016, Zeitler et al., 2022, Javurek et al., 11 May 2026, Kršek et al., 2024).

Several recurrent misconceptions are excluded by the cited results. A causal equation of state in GRHD guarantees hyperbolicity, not convexity; convexity additionally requires

$$\nabla_\mu J^\mu = 0, \qquad \nabla_\mu T^{\mu\nu}=0,$$03

Closedness of a preorder and local compactness do not guarantee causal convexity in the order-topological sense. Causal correlations are not restricted to convex mixtures of fixed total orders; dynamical orders are part of the polytope. Local consistency of causal marginals does not imply global realizability by a single SCM. Weighted-sum scalarization in causal sensitivity analysis recovers the full frontier only under the stated convexity and linearity conditions.

This suggests a common structural pattern rather than a common definition. Causality first restricts admissible dynamics, orders, correlations, models, or couplings. Convexity then supplies one of several forms of regularity: genuinely nonlinear wave structure, globalization of causal neighborhoods, facet-defining inequalities, linear-programming bounds, recoverable Pareto frontiers, or strong duality with attained optimizers. In that precise but plural sense, causal convexity is a cross-disciplinary label for the interaction between causal admissibility and convex geometry.