C-Capacity Monotonicity

• C-Capacity monotonicity is a structural property of non-additive set functions that ensures monotonic and extremal behavior under capacity constraints.
• It generalizes classical Choquet capacities and connects to optimal transport through capacity constraints, Möbius inversion, and combinatorial frameworks.
• Applications include constrained optimization, queueing theory, coding, and symplectic geometry, driving both theoretical advances and practical algorithms.

C-capacity monotonicity is a structural property arising in capacity theory, optimal transport, probabilistic potential theory, and related combinatorial frameworks. At its core, it describes a form of monotonic or extremal behavior for capacities—non-additive, monotone set functions—especially when considered in conjunction with additional constraints, such as capacity bounds, or within order-theoretic and probabilistic frameworks. The notion traces to and generalizes several fundamental objects: the classical Choquet capacities, the Möbius inversion theory on lattices, monotone and completely monotone set functions, and monotonicity concepts in constrained optimization problems such as capacity-constrained optimal transport.

1. Foundations: Capacities and Complete Monotonicity

A capacity, in the sense first developed by Choquet, is a monotone set function μ with μ(∅) = 0 and μ(X) = 1, defined on a finite set or a σ-algebra. In lattice-theoretic terms, a capacity p on a finite lattice L is a monotone function p : L → [0, 1] with p(0) = 0 and p(1) = 1. Complete monotonicity describes when all successive difference operators applied to p are nonnegative, that is, for any sequence $a_1,...,a_n$, $\Delta_{a_1,...,a_n} p(x) \ge 0.$ Such capacities admit a unique decomposition via their Möbius inverse, relating directly to probability measures on the lattice of dual order ideals. A key feature is that complete monotonicity is tightly bound to the positivity of the Möbius coefficients, which in turn dictate the “cumulative” or monotonic behavior of the capacity (Machida, 2011).

2. C-Capacity Monotonicity in Optimal Transport and Generalizations

In classical optimal transport, the c-cyclical monotonicity characterizes supports of optimal plans. In the presence of capacity constraints—where the transport density must satisfy $0 \leq h(x, y) \leq \bar{h}(x, y)$—the support of an optimizer is not c-cyclically monotone in the classical sense but rather exhibits c-capacity monotonicity (Chen, 28 Aug 2025).

Formally, a set $T \subset X \times Y$ is c-capacity monotone if, for any finitely supported probability measure $y$ on $T$ and any competitor $y'$ (sharing the same marginals and additional cost, as encoded by the dual additional cost $w_k(x, y)$), the cost is minimized:

$$\int_{X\times Y} c(x, y) \, dy(x, y) \le \int_{X\times Y} c(x, y) \, dy'(x, y).$$

When capacity constraints are absent (i.e., all additional dual costs $w_k$ vanish), this reduces to classical c-cyclical monotonicity. The definition explicitly incorporates the duality structure arising from capacity-constrained transport. The main result shows that optimizers are always supported on c-capacity monotone sets (Chen, 28 Aug 2025).

3. Combinatorial and Probabilistic Aspects

In finite and distributive lattice frameworks, capacities correspond to functions on order polytopes or Boolean lattices. C-capacity monotonicity here is reflected in the convex polytope structure: for $\mu: 2^N \to [0,1]$ with $\mu(\emptyset)=0$ and $\mu(N)=1$, monotonicity ($S \subset T \implies \mu(S) \leq \mu(T)$) ensures that the extreme points are ordered, and random generation techniques must respect this structure (Grabisch et al., 2022). The combinatorial machinery—especially Möbius inversion and Fréchet bounds—serves to analyze when the capacity or its increments (difference operators) satisfy local extremal properties, which is a discrete manifestation of c-capacity monotonicity (Machida, 2011).

Probabilistic interpretations reinforce this: a completely monotone capacity $p$ admits a representation as the marginal of a probability distribution on dual ideals. This intertwining underpins both the stochastic inequalities and the Fréchet-type bounds—maximal or minimal possible values compatible with the marginals.

4. Capacity Monotonicity in Analysis and Integration

C-capacity monotonicity has analytic consequences, particularly regarding the sublinearity and convergence properties of non-additive Choquet integrals (Ponce et al., 2023). Sublinearity of the Choquet integral

$\int (f + g)\, dH \leq \int f\, dH + \int g\, dH$

is equivalent to the “strong subadditivity” of the underlying capacity,

$H(E \cap F) + H(E \cup F) \leq H(E) + H(F).$

This equivalence is a strong form of capacity monotonicity, governing the interplay between set inclusion, union, and intersection. Such properties support analogues of Fatou's lemma and the dominated convergence theorem for non-additive measures, which are crucial for extending measure-theoretic convergence results to the capacitary setting.

The monotonicity of kernel-based capacities (e.g., the Riesz $p$-capacity) is also central. Riesz capacity is strictly decreasing in the kernel exponent $p$ when positive, interpolating between the set’s diameter (as $p \to -\infty$) and its volume (as $p \uparrow n$), and exhibits left and, under extra conditions, right continuity in $p$. The equilibrium measure’s dependence on $p$ further solidifies the analytic robustness of capacity monotonicity concepts (Clark et al., 16 Jun 2024).

5. Applications: Constraints, Control, and Optimization

C-capacity monotonicity arises directly in constrained optimization problems. For example, in principal-agent models with capacity constraints, Pareto optimal contracts are obtained by scaling the unconstrained optimal contract by a factor $a^*$ that is monotonic in the agent’s capacity; as the constraint relaxes, $a^*$ increases monotonically, restoring the unconstrained optimal incentive structure (Clark, 2 Dec 2024). This monotonic scaling relationship is a direct instantiation of capacity monotonicity: as capacity increases, the “distortion” due to the constraint vanishes monotonically.

In queueing theory, monotonicity of capacity functions (e.g., the Erlang C formula in the Halfin–Whitt regime) is rigorously established. The waiting probability decreases monotonically with system load, confirming conjectured monotonic behavior and providing conservative benchmarks for capacity planning (D'Auria, 2011).

In coding and information theory, channel capacity as a function of resource constraints is nondecreasing provided the channel law is static and cost functions are unbounded; this remains true in static interference channels and for joint optimization, ensuring that operational capacity never decreases with increased cost (Agrell, 2012).

6. Extensions: Geometry, Symplectic Capacities, and Network Calculus

Concepts analogous to c-capacity monotonicity appear in geometric analysis. In Carnot groups, monotonicity formulas built from right-invariant energies are valid, linking the sub-Riemannian geometry’s capacity-like functionals with monotonicity properties, though classical frequency monotonicity may fail (Garofalo, 2022). In symplectic geometry, cube-normalized capacities introduce a new form of monotonicity: agreement of all such normalized capacities on monotone toric domains, with explicit divergences in the non-monotone case (Gutt et al., 2022). This reflects a nuanced calibration of "capacity monotonicity" in geometric settings.

In wireless communications and stochastic network calculus, monotonicity properties govern cumulative and extreme-value capacities under various dependence structures (e.g., comonotonicity), directly affecting Quality-of-Service guarantees (Sun et al., 2015). Copula methods, MGFs, and Mellin transforms are deployed analytically to capture and exploit these monotonicity phenomena in the face of time-dependent statistical service variations.

7. Implications and Future Directions

The unifying thread of c-capacity monotonicity is its dual algebraic and analytic role: as a structural condition ensuring extremality under constraints (as with competitor measures in optimal transport), as a regularity property in set function theory, and as a monotonicity or calibration property in applied and geometric settings. Its generalizations prompt new challenges: extending duality and monotonicity concepts in non-additive and non-classical regimes, understanding the sharpness of monotonicity under various forms of dependence or symmetry, and developing efficient computational approaches consistent with these monotonicity structures.

The continued exploration of c-capacity monotonicity promises advances in both the theoretical detail of extremal measures and capacities and their robust application in constrained optimization settings, network systems, economics, and geometric analysis.