A model of anisotropic branched optimal transport

1. Problem Statement

The paper addresses the Anisotropic Branched Optimal Transport (ABOT) problem. This is a generalization of classical branched transport (which models efficient mass transport networks like irrigation or vascular systems) where the cost of transport depends not only on the amount of mass (multiplicity) but also on the spatial direction of the flow.

• Goal: Transport a source measure $\mu^-$ to a target measure $\mu^+$ (with equal total mass) via a transport path represented as a rectifiable 1-current $R$ with boundary $\partial R = \mu^+ - \mu^-$.
• Cost Functional: The cost is defined by the anisotropic $H$-mass, denoted $M_{H, \sigma_1}(R)$.
  • $H: \mathbb{R} \to [0, +\infty)$ is a "branching function" (even, lower semicontinuous, subadditive, $H(0)=0$) that penalizes separate transport, encouraging mass aggregation.
  • $\sigma_1$ is a 1-anisotropy, a continuous function on the Grassmannian $G(1, n)$ (directions) that assigns different costs to different spatial orientations.
• Objective: Prove the existence of a minimizer for the problem:
  $$\inf \{ M_{H, \sigma_1}(R) : R \in \mathcal{C} \}$$
  where $\mathcal{C}$ is the class of competitors satisfying the boundary condition.

2. Methodology and Framework

The authors employ the theory of rectifiable currents and tools from geometric measure theory.

A. Mathematical Definitions

• Currents: The transport path is modeled as a rectifiable 1-current $R = \mathbf{J}E, \tau, \theta\mathbf{K}$, where $E$ is a 1-rectifiable set, $\tau$ is a unit tangent vector, and $\theta$ is the multiplicity (flow intensity).
• Anisotropic Norm: For a vector $v$, the anisotropic norm is defined as $\|v\|_{\sigma_1} = \|v\|_\infty \sigma_1(\hat{v})$, where $\hat{v}$ is the direction of $v$.
• Relaxation: The authors define a sequential lower semicontinuous envelope $\Phi_{H, \sigma_k}$ of the mass functional with respect to the flat norm, starting from polyhedral currents.

B. Key Technical Tools

1. Integral Geometric Representations: A central concept is whether the anisotropy $\sigma_k$ admits an integral geometric representation. This means $\sigma_k(E)$ can be written as an average of projections over the Grassmannian:
   $$\sigma_k(E) = \int_{G(n-k, n)} [E, L] \, d\phi_{n-k}(L)$$
   where $[E, L]$ is a geometric factor related to the angle between subspaces.
2. Hypermetric Spaces: The existence of such representations is linked to the metric space $(\mathbb{R}^n, \|\cdot\|_{\sigma_1})$ being hypermetric. A space is hypermetric if a specific inequality involving distances and integer coefficients holds for any finite set of points.
3. Slicing Formulas: If an integral geometric representation exists, the anisotropic mass can be decomposed into a superposition of 0-dimensional masses (slices) along various directions. This allows the authors to reduce the lower semicontinuity of the complex functional to the known lower semicontinuity of the standard $H$-mass on 0-currents.

3. Key Contributions and Results

Theorem 1.7: Relaxation and Lower Semicontinuity

• Result: If the anisotropy $\sigma_k$ admits an integral geometric representation, then the anisotropic $H$-mass $M_{H, \sigma_k}$ coincides with its sequential lower semicontinuous envelope $\Phi_{H, \sigma_k}$ with respect to the flat norm.
• Significance: This establishes that the functional is lower semicontinuous, a crucial prerequisite for proving the existence of minimizers via the direct method in the calculus of variations.

Theorem 1.8: Existence of Minimizers

This is the main result of the paper. Under the assumptions that $H$ is non-decreasing, $\lim_{y\to 0^+} H(y)/y = +\infty$, and the infimum of the cost is finite:

1. Planar Case ($n=2$): A minimizer always exists.
   • Reasoning: In 2D, any convex 1-anisotropy admits an integral geometric representation (proven via approximation by polygons).
2. Higher Dimensions ($n \ge 3$): A minimizer exists if and only if the space $(\mathbb{R}^n, \|\cdot\|_{\sigma_1})$ is hypermetric.
   • Reasoning: The authors prove that for $k=1$, the existence of an integral geometric representation is equivalent to the hypermetricity of the normed space.

Technical Lemmas

• Compactness: The authors prove that minimizing sequences are relatively compact in the local flat topology within the class of acyclic currents (currents without cycles).
• $L^\infty$ Bound: They utilize a bound on the multiplicity $\theta$ for acyclic currents in 1D, which is essential for controlling the mass and ensuring the limit current remains rectifiable.

4. Significance and Implications

1. Generalization of Branched Transport: The paper successfully extends the classical isotropic branched transport theory (where cost depends only on multiplicity) to anisotropic settings. This is vital for applications where the medium is not uniform (e.g., transport in crystals, anisotropic materials, or biological tissues with preferred directions).
2. Geometric Characterization: The paper provides a sharp geometric characterization of when existence holds in higher dimensions. It links the analytical problem of transport to the geometric property of hypermetricity.
   • Example: The $\ell_1$ and $\ell_2$ norms are hypermetric, so transport with these costs admits minimizers. The $\ell_\infty$ norm (in $n \ge 3$) is not hypermetric, implying that for certain anisotropic costs, minimizers might not exist or the problem requires different formulations.
3. Dimensional Distinction: The work highlights a fundamental difference between the planar case ($n=2$) and higher dimensions. In 2D, convexity of the anisotropy is sufficient for existence. In $n \ge 3$, convexity is not enough; the stronger hypermetric condition is required.
4. Open Problems: The authors note that for $k$-currents with $k \in \{2, \dots, n-1\}$, a necessary and sufficient condition for integral geometric representation is not yet known, leaving an open avenue for future research in higher-dimensional geometric measure theory.

Summary

Bellettini and Marchese establish the existence of optimal transport networks in anisotropic environments by proving that the anisotropic mass functional is lower semicontinuous and that minimizing sequences are compact. Their work demonstrates that in dimensions $n \ge 3$, the solvability of the problem is strictly tied to the hypermetric nature of the underlying anisotropic norm, providing a rigorous bridge between optimal transport theory and the geometry of normed spaces.