---
title: Lower-Dimensional Anisotropic Minkowski Content
url: https://www.emergentmind.com/topics/lower-dimensional-anisotropic-minkowski-content
type: topic
---

# Lower-Dimensional Anisotropic Minkowski Content

Lower-dimensional anisotropic Minkowski content is the study of small-scale volume growth of anisotropic parallel sets \(S\oplus rC\) when the underlying set \(S\subset \mathbb R^n\) is expected to behave like a \(k\)-dimensional object, with \(1\le k\le n-1\). The Euclidean unit ball is replaced by a convex body \(C\), and the leading coefficient of \(\lambda^n(S\oplus rC)\) at order \(r^{n-k}\) is interpreted as a \(C\)-dependent lower-dimensional size. In the contemporary literature, the subject splits into three closely related regimes: codimension-one boundary growth and anisotropic perimeter, arbitrary-codimension rectifiable sets, and fractal or averaged curvature-direction theories based on parallel sets [1203.5190], [2601.22681], [1211.3421].

## 1. Definitions and geometric framework

The anisotropy is encoded by a convex body \(C\). In the codimension-one BV literature one usually assumes \(C\in\mathcal C_0^n\), meaning a nonempty compact convex set with \(0\in\operatorname{int}(C)\), and uses its support function
\[
h_C(y)\coloneq \sup_{x\in C} x\cdot y,
\]
together with the polar gauge
\[
h_C^\circ(x)\coloneq \sup_{\substack{y\in\mathbb R^n\\ h_C(y)\le 1}} x\cdot y.
\]
The anisotropic parallel set is \(E_{r,C}=E\oplus rC\), equivalently
\[
E_{r,C}=\{x\in\mathbb R^n:\operatorname{dist}_C(x,E)\le r\},
\qquad
\operatorname{dist}_C(x,E)=\inf_{y\in E} h_C^\circ(x-y).
\]
This is the basic neighborhood operator from which anisotropic contents are extracted [1203.5190], [2509.06545].

For general lower-dimensional content, a standard normalized definition is
\[
\mathcal M^s_{r,C}(E)\coloneq \frac{\lambda^n(E\oplus rC)}{\omega_{n-s}r^{\,n-s}},
\qquad r>0,
\]
with lower and upper limits \(\mathcal M_C^s(E)_*\) and \(\mathcal M_C^s(E)^*\), and full content \(\mathcal M_C^s(E)\) when the limit exists. In the main rectifiable theory, the dimension of interest is \(s=k\), so one studies
\[
\mathcal M_C^k(S)=\lim_{r\to0_+}\frac{\lambda^n(S\oplus rC)}{\omega_{n-k}r^{\,n-k}}
\]
whenever the limit exists [2601.22681]. A parallel but unnormalized convention also appears:
\[
\mathcal M^s_{r,C}(E)\coloneq \frac{V_{E,C}(r)}{r^{\,n-s}},
\qquad
V_{E,C}(r)=\lambda^n(E\oplus rC),
\]
with corresponding outer version
\[
\mathcal{SM}^s_{r,C}(E)\coloneq \frac{V_{E,C}(r)-V_{E,C}(0)}{r^{\,n-s}}.
\]
The absence of the Euclidean factor \(\omega_{n-s}\) is deliberate there: for general \(C\) there is no canonical anisotropic analogue of the Euclidean normalization [2509.06545].

A further nonclassical variant allows the structuring element itself to be lower-dimensional. If \(Q\subset \mathbb R^n\) is compact convex with \(\dim Q=k<n\), then \(A+rQ\) thickens only along the directions in \(L=\operatorname{span}(Q)\). In that setting one defines
\[
M_Q(E):=\lim_{r\to 0+}\frac{\lambda_n(E+rQ)}{2r},
\qquad
SM_Q(A):=\lim_{r\to 0+}\frac{\lambda_n((A+rQ)\setminus A)}{r},
\]
provided the limits exist. This regime is anisotropic not only because of directional weighting, but because the neighborhood operator probes only a proper subspace of directions [2504.03339].

## 2. Codimension-one theory and anisotropic perimeter

The earliest rigorous anisotropic identifications concern codimension one. For a set \(E\) of finite perimeter in \(\Omega\), with reduced boundary \(\partial^*E\) and measure-theoretic outer normal \(\nu_E\), the anisotropic perimeter is
\[
h_C(E;\Omega)
=
\int_{\partial^*E\cap\Omega} h_C(\nu_E(x))\,d\mathcal H^{n-1}(x),
\]
and the anisotropic outer Minkowski content is obtained from first-order growth of \(E+\varepsilon C\). The central result is that, under essentially the same hypotheses under which the Euclidean outer Minkowski content equals perimeter, one has
\[
\lim_{\varepsilon\to0}
\frac{|\Omega\cap(E+\varepsilon C)|-|E|}{\varepsilon}
=
\int_{\partial^*E\cap\Omega} h_C(\nu_E)\,d\mathcal H^{n-1},
\]
so the first-order anisotropic outer volume growth detects the codimension-one anisotropic perimeter density \(h_C(\nu_E)\,d\mathcal H^{n-1}\) [1203.5190].

This codimension-one theory is explicitly not a theory for \(k\)-dimensional sets with \(k<n-1\). Its “lower-dimensional” character is only the standard tubular one: dividing the \(n\)-dimensional volume increment by \(\varepsilon\) extracts an \((n-1)\)-dimensional coefficient. In the isotropic case \(C=B(0,1)\), the density reduces to \(1\), and the limit recovers \(\operatorname{Per}(E;\Omega)=\mathcal H^{n-1}(\partial^*E\cap\Omega)\) [1203.5190].

A complementary codimension-one development concerns the Minkowski content of the topological boundary. For a set of finite perimeter \(E\), the \(C\)-anisotropic Minkowski content of \(\partial E\) is the two-sided quantity
\[
\mathcal M_C(\partial E;\Omega)
=
\lim_{\varepsilon\to 0^+}
\frac{1}{2\varepsilon}\lambda^n\Big(\big((\partial E\cap\Omega)\oplus \varepsilon C\big)\cap \Omega\Big),
\]
when the limit exists. The decisive equivalence theorem states that
\[
\mathcal M_C(\partial E;\Omega)
=
\frac12\Big(\operatorname{Per}_{h_C}(E;\Omega)+\operatorname{Per}_{h_C}(\Omega\setminus E;\Omega)\Big)
\]
for one convex body \(C\in\mathcal C_0^n\) if and only if the corresponding formula holds for every other \(C'\in\mathcal C_0^n\); in particular, the isotropic and anisotropic codimension-one formulas stand or fall together [2508.08156].

The arithmetic mean above is geometrically forced. Two-sided boundary thickening counts both sides of the hypersurface, so the density is
\[
\frac12\big(h_C(\nu_E)+h_C(-\nu_E)\big).
\]
If \(C\) is not centrally symmetric, then \(h_C(\nu)\neq h_C(-\nu)\) in general, and
\[
\operatorname{Per}_{h_C}(E;\Omega)
=
\int_{\partial^*E\cap\Omega} h_C(\nu_E)\,d\mathcal H^{n-1}
\]
need not equal the anisotropic perimeter of the complement. This asymmetry is a defining feature of nonsymmetric anisotropy [1203.5190], [2508.08156].

## 3. Arbitrary codimension for rectifiable sets

A genuine arbitrary-codimension theory was established for compact \(k\)-rectifiable and countably \(\mathcal H^k\)-rectifiable sets. If \(S\subset\mathbb R^n\) is countably \(\mathcal H^k\)-rectifiable, then for \(\mathcal H^k\)-a.e. \(x\in S\) there is an approximate tangent plane \(T_x^kS\), normal space \(N_x^{n-k}S=(T_x^kS)^\perp\), and projected anisotropy
\[
C_x\coloneq P_{(T_x^kS)^\perp}(C)=P_{N_x^{n-k}S}(C).
\]
The main limiting functional is
\[
\Phi_S(C)
\coloneq
\frac{1}{\omega_{n-k}}
\int_S \mathcal H^{n-k}(C_x)\,d\mathcal H^k(x),
\]
and for a \(k\)-rectifiable compact set one has the exact identity
\[
\mathcal M_C^k(S)
=
\frac{1}{\omega_{n-k}}
\int_S \mathcal H^{n-k}(C_x)\,d\mathcal H^k(x)
\]
for every convex body \(C\) [2601.22681].

This formula identifies the correct anisotropic density in arbitrary codimension: not a support function, but the \((n-k)\)-dimensional volume of the orthogonal projection of \(C\) onto the approximate normal space. The geometric mechanism is local flattening. Near a rectifiable point \(x\), the set looks like its tangent \(k\)-plane, so \(S\oplus rC\) is asymptotically governed only by the part of \(C\) visible in the normal directions. Tangential components of \(C\) do not contribute to the leading \(r^{n-k}\)-term [2601.22681].

The same paper extends the formula to countably \(\mathcal H^k\)-rectifiable compact sets under an AFP-type hypothesis. In the full-dimensional case \(C\in\mathcal C^{n,n}\), the relevant condition is the AFP-\(k\)-condition. For lower-dimensional anisotropy, when \(C\subseteq L\) and \(\dim L=m\), the generalized condition is the AFP-\(k\)-condition relative to \(L\):
\[
\mu(B(x,r))
\ge \gamma\, r^{k+m-n}\,
\mathcal H^{n-m}\!\big(P_{L^\perp}(S\cap B(x,r))\big),
\qquad x\in S,\ r\in(0,1),
\]
for some \(\gamma>0\) and some Radon measure \(\mu\ll\mathcal H^k\). If \(k+m-n>0\), this again yields
\[
\mathcal M_C^k(S)
=
\frac{1}{\omega_{n-k}}
\int_S \mathcal H^{n-k}(C_x)\,d\mathcal H^k(x)
\]
for every \(C\subseteq L\) [2601.22681].

Two threshold cases are completely described. If \(k+m=n\), no AFP assumption is needed and there is a slicing formula
\[
\mathcal M_C^k(S)
=
\frac{\mathcal H^{n-k}(C)}{\omega_{n-k}}
\int_{L^\perp}\mathcal H^0(S\cap(x+L))\,d\mathcal H^k(x),
\]
which reduces to the same tangent-plane formula for countably rectifiable \(S\). If \(k+m<n\), then
\[
\mathcal M_C^k(S)=0,
\]
because the projected anisotropy has dimension \(<n-k\), so \(\mathcal H^{n-k}(C_x)=0\) almost everywhere [2601.22681].

The codimension-one specialization recovers the familiar support-function density. When \(k=n-1\), the normal space is one-dimensional and
\[
\mathcal H^1(C_x)=\frac12\big(h_C(\nu_S(x))+h_C(-\nu_S(x))\big),
\]
hence
\[
\mathcal M_C^{n-1}(S)
=
\frac12\int_S\big(h_C(\nu_S)+h_C(-\nu_S)\big)\,d\mathcal H^{n-1}.
\]
For \(C=B(0,1)\), the projected normal section is the Euclidean unit ball in the normal space, so \(\mathcal H^{n-k}(C_x)=\omega_{n-k}\), and the formula reduces to Federer’s isotropic identification \(\mathcal M_{B(0,1)}^k(S)=\mathcal H^k(S)\) [2601.22681].

## 4. Lower-dimensional structuring elements and relative AFP conditions

A distinct but closely related theory fixes the measured set at codimension one and lowers the dimension of the structuring element. Let \(E\subset\mathbb R^n\) be compact and countably \(\mathcal H^{n-1}\)-rectifiable, and let \(Q\subset L\) be a nonempty compact convex set with \(\dim L=k<n\). The key sufficient hypothesis is the AFP-condition relative to \(L\):
\[
\nu(B(x,r))
\ge
\gamma\, r^{k-1}\,\lambda_{n-k}\!\big(P_{L^\perp}(E\cap B(x,r))\big),
\qquad x\in E,\ r\in(0,1),
\]
for some Radon measure \(\nu\ll\mathcal H^{n-1}\) and \(\gamma>0\). Under this weaker, directionally adapted condition, the \(Q\)-Minkowski content exists and satisfies
\[
M_Q(E)=\int_E h_{\diamond Q}(v_E(x))\,\mathcal H^{n-1}(dx),
\qquad
\diamond Q=\frac{Q+(-Q)}2,
\]
where \(v_E(x)\) is a unit normal defined up to sign. The symmetral \(\diamond Q\) appears because \(E+rQ\) expands on both sides of the hypersurface, so only the symmetric width matters [2504.03339].

This relative AFP condition is strictly weaker than the classical Ambrosio–Fusco–Pallara density condition when \(k<n\). The right-hand side measures only the transverse \((n-k)\)-dimensional spread of \(E\) through \(P_{L^\perp}\), together with the expected \(r^{k-1}\) behavior along the \(L\)-directions. Accordingly, lower-dimensional thickening can succeed even when isotropic thickening fails [2504.03339].

The corresponding outer theory identifies one-sided growth with anisotropic perimeter. For a set \(A\) of finite perimeter,
\[
P_Q(A)=\int_{S^{n-1}} h_{Q\cup\{0\}}(v)\,S_{n-1}^*(A,dv),
\]
and if \(\partial A\) admits the \(Q\)-Minkowski content with
\[
M_Q(\partial A)=P_{\diamond Q}(A),
\]
then \(A\) admits outer \(Q\)-Minkowski content and
\[
SM_Q(A)=P_Q(A).
\]
A practical criterion is obtained when \(\partial A\) is compact, satisfies the AFP-condition relative to \(L\), and
\[
P(A)=\mathcal H^{n-1}(\partial A);
\]
then \(SM_Q(A)=P_Q(A)\) [2504.03339].

A particularly sharp one-dimensional directed case is \(Q=[0,u]\), \(u\in S^{n-1}\). Then
\[
SM_{[0,u]}(A)=\int_{S^{n-1}} (u,v)_+\, S_{n-1}^*(A,dv),
\]
and the result holds for every finite-perimeter set after choosing a suitable representative of its \(L^1\)-class. This is exceptional in that no AFP-type assumption is required [2504.03339].

The geometric distinction from isotropic thickening is visible in explicit examples. One example constructs a compact finite-perimeter set \(A\subset\mathbb R^3\) such that isotropic outer Minkowski content does not exist, while \(SM_{B^2}(A)\) exists for every two-dimensional unit disk \(B^2\subset\mathbb R^3\). Another product example gives \(A=C\times D\) with pathological isotropic outer growth in the complementary factor, yet existence of outer content for \(Q=B^k\times\{0\}\). These examples show that lower-dimensional thickening is not a degenerate version of the full-dimensional theory; it probes different geometry and admits weaker existence criteria [2504.03339].

## 5. Anisotropic \(\mathcal S\)-content, Kneser functions, and dimensions

A further development compares anisotropic lower-dimensional Minkowski content with anisotropic \(\mathcal S\)-content, the latter being defined through anisotropic perimeter of parallel sets. For compact \(E\subset\mathbb R^n\),
\[
S_{E,C}(r)\coloneq \operatorname{Per}_{h_C}(E_{r,C};\mathbb R^n),
\]
and away from an at most countable exceptional set,
\[
(V_{E,C})'(r)=S_{E,C}(r).
\]
The anisotropic \(\mathcal S\)-content is then
\[
\mathcal S_C^s(E)_*
=
\liminf_{r\to0_+}\frac{S_{E,C}(r)}{(n-s)r^{\,n-s-1}},
\qquad
\mathcal S_C^s(E)^*
=
\limsup_{r\to0_+}\frac{S_{E,C}(r)}{(n-s)r^{\,n-s-1}},
\]
with the usual convention when lower and upper values agree [2509.06545].

The structural theorem behind the comparison is that
\[
V_{E,C}(r)=\lambda^n(E\oplus rC)
\]
is a Kneser function of order \(n\) for every compact \(E\) and every \(C\in\mathcal C_0^n\). This extends the classical Euclidean result and the symmetric-convex case to arbitrary convex bodies with \(0\in\operatorname{int}(C)\). The proof uses the monotonicity of
\[
\kappa_{E,C}(r)=\frac{S_{E,C}(r)}{r^{n-1}},
\]
which is decreasing in \(r\) [2509.06545].

For \(\lambda^n(E)=0\), the Kneser framework yields the basic anisotropic inequalities
\[
\mathcal S_C^s(E)_*
\le
\mathcal M_C^s(E)_*
\le
\mathcal M_C^s(E)^*
\le
\mathcal S_C^s(E)^*.
\]
There is also a sharper upper comparison,
\[
\mathcal M_C^s(E)^*
\ge
\frac{n-s}{n}\,\mathcal S_C^s(E)^*,
\]
and a lower isoperimetric comparison with shifted dimension parameter,
\[
c(C,n,s)\,\mathcal S_C^{\,s\frac{n-1}{n}}(E)_*
\ge
\big(\mathcal M_C^s(E)_*\big)^{\frac{n-1}{n}}.
\]
Thus the asymmetry between upper and lower theories survives intact in the anisotropic setting: upper contents are tightly coupled, while lower contents admit only weaker, dimension-shifting control [2509.06545].

The corresponding dimensions reflect the same pattern. The upper anisotropic Minkowski and \(\mathcal S\)-dimensions coincide,
\[
\dim_{\mathcal M}(E)^*=\dim_{\mathcal S}^C(E)^*,
\]
and the upper \(\mathcal S\)-dimension is independent of \(C\). For lower dimensions one has only
\[
\dim_{\mathcal S}^C(E)_*
\ge
\frac{n-1}{n}\,\dim_{\mathcal M}(E)_*.
\]
This mirrors the earlier isotropic theory, where explicit examples show that lower surface-based and lower volume-based dimensions can separate sharply [1003.3776], [2509.06545].

The planar Sierpiński gasket gives a concrete anisotropic example. For \(D=\log_2 3\), the \(D\)-dimensional \(C\)-anisotropic Minkowski content and anisotropic \(\mathcal S\)-content do not exist for any \(C\in\mathcal C_0^2\); rather,
\[
\mathcal S_C^D(E)_*
<
\mathcal M_C^D(E)_*
<
\mathcal M_C^D(E)^*
<
\mathcal S_C^D(E)^*.
\]
Here the anisotropy enters through the support-function sum
\[
u_2=h_C(v_1)+h_C(v_2)+h_C(v_3),
\]
where \(v_1,v_2,v_3\) are the outward normals to the sides of the generating equilateral triangle. The example shows that even for a highly symmetric self-similar set, lower and upper anisotropic contents need not collapse to a single value [2509.06545].

## 6. Fractal curvature-direction measures, dependence on \(C\), and scope

For self-conformal sets, the lower-dimensional anisotropic picture becomes measure-valued and typically averaged. If \(F\subset\mathbb R^d\) is self-conformal with dimension \(D\), the \(k\)-th fractal curvature-direction measure is defined, in Cesàro form, by
\[
C_k^{\mathrm{frac}}(F,\cdot)
=
\lim_{\varepsilon\searrow 0}
\frac{1}{|\ln \varepsilon|}
\int_\varepsilon^1
r^{D-k} C_k(F_r,\cdot)\,\frac{dr}{r},
\qquad k=0,\dots,d,
\]
as a measure on \(\mathbb R^d\times S^{d-1}\). For \(k=d\) this is averaged Minkowski content, for \(k=d-1\) averaged surface content, and for \(k<d-1\) it yields lower-order, lower-dimensional contents extracted from parallel sets. Because the measures live on \(\mathbb R^d\times S^{d-1}\), they are genuinely directional and hence anisotropic [1211.3421].

The self-conformal theory differs from the rectifiable one in two ways. First, convergence is generally only proved in logarithmic Cesàro average, not pointwise in \(r\to0\), because the raw rescaled quantities may oscillate. Second, the limiting measures are not merely scalar densities but curvature-direction measures involving normal directions and distortion limits from the conformal iterated function system. Existence is proved under the Open Set Condition, regularity assumptions for parallel sets, and, for \(k<d-1\), nontrivial uniform integrability conditions controlling curvature on overlap regions [1211.3421].

Dependence on the anisotropy also bifurcates according to whether \(C\) is full-dimensional. For countably \(\mathcal H^k\)-rectifiable compact sets and full-dimensional convex bodies \(C,C'\in\mathcal C^{n,n}\), existence with the geometric value for one \(C\) is equivalent to existence with the corresponding value for every other full-dimensional \(C'\). By contrast, when \(C\) is lower-dimensional, dependence on \(C\) can be genuine because the span \(L=\operatorname{span}(C)\) selects the directions along which the set is probed, and the relevant AFP condition must be formulated relative to \(L\) [2601.22681].

Several limitations remain intrinsic to the subject. Codimension-one results such as the outer-content theorem for finite-perimeter sets do not address \(k<n-1\) directly [1203.5190]. The transfer principle for boundary Minkowski content is also codimension-one [2508.08156]. The theory with lower-dimensional structuring elements provides strong sufficient conditions, not a complete characterization of when
\[
SM_Q(A)=P_Q(A)
\]
holds in full generality [2504.03339]. The fractal self-conformal theory is predominantly an averaged theory rather than an ordinary pointwise one [1211.3421]. These caveats mark the present boundary of the subject.

Taken together, the modern theory identifies a coherent geometric principle. For codimension one, anisotropic tube growth is governed by support-function weights \(h_C(\nu)\) or their symmetrized version. In arbitrary codimension, the correct density is the \((n-k)\)-dimensional volume of the projection of \(C\) onto the approximate normal space,
\[
\mathcal H^{n-k}\!\big(P_{(T_x^kS)^\perp}(C)\big).
\]
For fractal sets, the same parallel-set philosophy persists, but the relevant limits may be directional curvature measures and may exist only after Cesàro averaging. This is the current mathematical content of lower-dimensional anisotropic Minkowski content across rectifiable, BV, and self-conformal settings [2601.22681], [1211.3421].

Source: https://www.emergentmind.com/topics/lower-dimensional-anisotropic-minkowski-content