Rectifiability via Slicing: Criteria & Advances

• The paper demonstrates that if slicing along a positive measure set of m-planes yields almost-atomic disintegration, then m-rectifiability of the measure is ensured.
• It introduces an advanced Carathéodory slicing approach with L^p control to decompose measures into rectifiable and purely unrectifiable parts.
• The work resolves Federer's integralgeometric problem for p>1 and offers unified rectifiability results for variational and jump set applications.

Rectifiability via slicing refers to a set of geometric conditions—expressed in terms of slicing and disintegration of measures—that guarantee a measure in Euclidean (or more general) space is carried by a countably rectifiable set of a given dimension. Slicing-based criteria provide both necessary and sufficient structural characterizations for the rectifiability of Radon measures and are foundational in the analysis of geometric measure theory, analysis of jump sets in variational problems, and the study of currents and their generalizations. The modern approach extends the classical framework by leveraging integralgeometric measures constructed via Carathéodory’s method and $L^p$-control over slice families, allowing resolution of longstanding open questions and novel results even in situations where chain or current structure is unavailable.

1. Definition of Integralgeometric Measures and Slicing

Given integers $0\le m\le n$ and $1\le p\le\infty$, and a "transversal" parameter set $A$ with Lebesgue measure, one defines a family of maps $P_a:\mathbb{R}^n\to\mathbb{R}^m$ parameterized by $a\in A$ and a measurable family of Borel-regular measures $\{\mu_a\}_{a\in A}$ on $\mathbb{R}^n$. The $L^p$-slice content of a Borel set $B\subset\mathbb{R}^n$ is

$$S_p(B) := \left(\int_A [\mu_a(B)]^p \, da\right)^{1/p}$$

(with the natural modification for $p=\infty$). The associated outer measure is given by the Carathéodory construction: $$I_{m,p}(E) := \sup_{\delta>0} \inf\left\{\sum_{i}S_p(B_i):\ E\subset\bigcup_i B_i,\ \operatorname{diam} B_i \le \delta\right\}.$$ This generalizes the classical $p$-integralgeometric measures $\mathrm{IG}_{m,p}$ of Federer. When $A=\mathrm{Gr}(n,m)$, the Grassmannian, and $P_V$ is orthogonal projection onto $V$, one reproduces the integralgeometric content $$\|V\mapsto \mathcal{H}^m(P_V(E))\|_{L^p(\mathrm{Gr}(n,m))}$$.

A Radon measure $\mu$ on $\mathbb{R}^n$ can be "sliced" along $m$-planes via push-forward and a disintegration theorem: For a Lipschitz map $T_V$, there exist conditional measures $\mu_{V,y}$ supported on fibres $T_V^{-1}(y)$ so that $\mu = \int_V \mu_{V,y} \, d(T_{V\#}\mu)(y)$, with $T_{V\#}\mu\ll \mathcal{H}^m$. The family $\{\mu_{V,y}\}$ is termed the slicing of $\mu$ along $V$.

2. Slicing-Based Rectifiability Criterion: The Main Theorem

Let $\mu$ be a Radon measure in $\mathbb{R}^n$ and $0\le m\le n$. The slicing-based rectifiability criterion asserts: If there is a measurable set $A\subset\mathrm{Gr}(n,m)$ of positive Haar measure such that for all $V\in A$,

• (a) the push-forward $T_{V\#}\mu\ll \mathcal{H}^m$,
• (b) for $T_{V\#}\mu$-almost every $y\in V$, the slice $\mu_{V,y}$ is a finite sum of Dirac masses,

then $\mu$ is $m$-rectifiable (i.e., supported on a countably $m$-rectifiable set $R\subset\mathbb{R}^n$ with $\mu(\mathbb{R}^n\setminus R)=0$). Furthermore, if each $T_V$ is locally $\alpha$-Hölder, then $\mu\ll \mathcal{H}^m\lfloor R$.

These hypotheses are sharp: Mattila's classical counterexample shows that if (a)–(b) only hold on a countable dense set of planes, rectifiability may fail (Tasso, 2022).

3. Structure of the Proof and New Technical Contributions

Classical approaches, including White's rectifiable-slices theorem for flat chains and Ambrosio–Kirchheim’s extension to metric currents, relied on boundary and current structures, mass bounds, and iterative slicing with induction. The present criterion operates without such algebraic or geometric structure and covers arbitrary Radon measures.

The new technical core consists of:

• A structure theorem for measures with vanishingly small cone mass in almost every parameterized slice, proving that such measures decompose into rectifiable and purely unrectifiable components. This is established via a novel covering and projection argument (Theorem 3.8).
• Double use of disintegration: the measure is first lifted to a product space, then re-sliced along fibres.
• Uniform $L^p$-control of normalized covering densities, relying on weak compactness in $L^p(A)$ for $p>1$ (Proposition 4.8), to obtain almost-atomic disintegration and preclude pathological concentration. Reflexivity of $L^p$ is essential here, which distinguishes the $p=1$ case.
• Reduction techniques, relating vanishing of the measure on product sets to vanishing for almost every parameter (Proposition 4.10), and identification of the Radon–Nikodym derivative as a slicing-density function (Lemma 6.1).
• Synthesis of all ingredients into a rectifiable restriction for $\mu$ (Theorem 6.2).

4. Applications: Federer's Integralgeometric Problem and the Jump Set Slicing Theorem

Federer questioned whether finite $p$-integralgeometric measure ($\mathrm{IG}_{m,p}(E)<\infty$) implies $m$-rectifiability of $\mathrm{IG}_{m,p}\lfloor E$. For $p=\infty$, the answer is known to be affirmative; for $p=1$, Mattila’s example shows failure. The present slicing criterion resolves this problem for all $p>1$: if $1<p\le\infty$ and $\mathrm{IG}_{m,p}(E)<\infty$, then $$\mathrm{IG}_{m,p}\lfloor E=\theta(x)\,\mathcal{H}^m\lfloor R$$ for some countably $m$-rectifiable set $R$ and function $\theta\in L^1_{\mathrm{loc}}(R,\mathcal{H}^m)$ (Tasso, 2022).

For measures defined via slicing of jump sets (e.g., in $BD$-type function spaces), the slicing-based criterion provides a unified proof of rectifiability for jump sets under $L^p$-finite slicing content and a suitable oscillation control hypothesis, both for linear and curvilinear slices. The approach bypasses classical codimension-one and parallelogram-law techniques (Almi et al., 2022).

5. Key Intermediate Results

Several technical lemmas underpin the main theorem. Principal among these are:

• Theorem 2.3: Disintegration of Radon measures along general Borel maps.
• Theorem 3.8: Decomposition for measures admitting few large cones in slices.
• Proposition 4.8: Weak compactness of normalized covering densities in $L^p$, exploiting reflexivity for $p>1$.
• Proposition 4.10: Reduction of zero-mass conditions on product sets.
• Lemma 6.1: Expression of the Radon–Nikodym derivative as a limit of local normalized slice masses.
• Theorem 6.2: Construction of the $m$-rectifiable carrier set.

All steps are interlinked: $L^p$-control precludes singular blow-ups, and the double disintegration argument precisely localizes rectifiable structure.

6. Extensions, Critical Parameters, and Limitations

For $p>1$, the Carathéodory–slicing approach yields global rectifiability results and identification of the absolutely continuous part with respect to $\mathcal{H}^m$, concurring with the classical theory on rectifiable sets. For $p=1$, only a decomposition into rectifiable and singular parts is possible; the singular part corresponds precisely to measures supported on sets whose slices "almost always" miss rectifiable structure, exemplified by Mattila’s example.

A further extension allows the projections $P_a$ to be locally $\alpha$–Hölder or even curvilinear (in Riemannian or more general settings). This enables slicing-based rectifiability criteria for very general function spaces, such as those of generalized bounded deformation, even on manifolds (Almi et al., 2022).

7. Impact and Connections to Broader Geometric Analysis

The slicing-based rectifiability criterion represents a fundamental advance in the geometric measure-theoretic understanding of measure structure, bypassing strong algebraic structure and facilitating extension to variational problems, jump sets, and integralgeometric analysis. It accesses rectifiability in arbitrary Radon measures and clarifies the relationship between integralgeometric measure, $L^p$-integrability, and rectifiability. A plausible implication is that similar duality and disintegration techniques might inform future developments in metric measure spaces, object recognition in metric geometry, and variational problems on irregular domains.

Key references: (Tasso, 2022, Almi et al., 2022).