Lower-Dimensional Anisotropic Minkowski Content

• Lower-dimensional anisotropic Minkowski content is a measure that quantifies the r^(n-k) order volume growth of parallel sets using a convex body as the structuring element.
• The methodology replaces the Euclidean ball with an anisotropic convex body, employing support functions and projections onto normal spaces to capture k-rectifiable geometries.
• The framework has significant implications for understanding anisotropic perimeters, fractal curvature-direction measures, and the interplay between BV theory and self-conformal analysis.

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 (Chambolle et al., 2012, Fryš, 30 Jan 2026, Bohl, 2012).

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 $S\subset \mathbb R^n$0, and uses its support function

$S\subset \mathbb R^n$1

together with the polar gauge

$S\subset \mathbb R^n$2

The anisotropic parallel set is $S\subset \mathbb R^n$3, equivalently

$S\subset \mathbb R^n$4

This is the basic neighborhood operator from which anisotropic contents are extracted (Chambolle et al., 2012, Fryš, 8 Sep 2025).

For general lower-dimensional content, a standard normalized definition is

$S\subset \mathbb R^n$5

with lower and upper limits $S\subset \mathbb R^n$6 and $S\subset \mathbb R^n$7, and full content $S\subset \mathbb R^n$8 when the limit exists. In the main rectifiable theory, the dimension of interest is $S\subset \mathbb R^n$9, so one studies

$k$0

whenever the limit exists (Fryš, 30 Jan 2026). A parallel but unnormalized convention also appears: $k$1 with corresponding outer version

$k$2

The absence of the Euclidean factor $k$3 is deliberate there: for general $k$4 there is no canonical anisotropic analogue of the Euclidean normalization (Fryš, 8 Sep 2025).

A further nonclassical variant allows the structuring element itself to be lower-dimensional. If $k$5 is compact convex with $k$6, then $k$7 thickens only along the directions in $k$8. In that setting one defines

$k$9

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 (Kiderlen et al., 4 Apr 2025).

2. Codimension-one theory and anisotropic perimeter

The earliest rigorous anisotropic identifications concern codimension one. For a set $1\le k\le n-1$0 of finite perimeter in $1\le k\le n-1$1, with reduced boundary $1\le k\le n-1$2 and measure-theoretic outer normal $1\le k\le n-1$3, the anisotropic perimeter is

$1\le k\le n-1$4

and the anisotropic outer Minkowski content is obtained from first-order growth of $1\le k\le n-1$5. The central result is that, under essentially the same hypotheses under which the Euclidean outer Minkowski content equals perimeter, one has

$1\le k\le n-1$6

so the first-order anisotropic outer volume growth detects the codimension-one anisotropic perimeter density $1\le k\le n-1$7 (Chambolle et al., 2012).

This codimension-one theory is explicitly not a theory for $1\le k\le n-1$8-dimensional sets with $1\le k\le n-1$9. Its “lower-dimensional” character is only the standard tubular one: dividing the $C$0-dimensional volume increment by $C$1 extracts an $C$2-dimensional coefficient. In the isotropic case $C$3, the density reduces to $C$4, and the limit recovers $C$5 (Chambolle et al., 2012).

A complementary codimension-one development concerns the Minkowski content of the topological boundary. For a set of finite perimeter $C$6, the $C$7-anisotropic Minkowski content of $C$8 is the two-sided quantity

$C$9

when the limit exists. The decisive equivalence theorem states that

$\lambda^n(S\oplus rC)$0

for one convex body $\lambda^n(S\oplus rC)$1 if and only if the corresponding formula holds for every other $\lambda^n(S\oplus rC)$2; in particular, the isotropic and anisotropic codimension-one formulas stand or fall together (Fryš, 11 Aug 2025).

The arithmetic mean above is geometrically forced. Two-sided boundary thickening counts both sides of the hypersurface, so the density is

$\lambda^n(S\oplus rC)$3

If $\lambda^n(S\oplus rC)$4 is not centrally symmetric, then $\lambda^n(S\oplus rC)$5 in general, and

$\lambda^n(S\oplus rC)$6

need not equal the anisotropic perimeter of the complement. This asymmetry is a defining feature of nonsymmetric anisotropy (Chambolle et al., 2012, Fryš, 11 Aug 2025).

3. Arbitrary codimension for rectifiable sets

A genuine arbitrary-codimension theory was established for compact $\lambda^n(S\oplus rC)$7-rectifiable and countably $\lambda^n(S\oplus rC)$8-rectifiable sets. If $\lambda^n(S\oplus rC)$9 is countably $r^{n-k}$0-rectifiable, then for $r^{n-k}$1-a.e. $r^{n-k}$2 there is an approximate tangent plane $r^{n-k}$3, normal space $r^{n-k}$4, and projected anisotropy

$r^{n-k}$5

The main limiting functional is

$r^{n-k}$6

and for a $r^{n-k}$7-rectifiable compact set one has the exact identity

$r^{n-k}$8

for every convex body $r^{n-k}$9 (Fryš, 30 Jan 2026).

This formula identifies the correct anisotropic density in arbitrary codimension: not a support function, but the $C$0-dimensional volume of the orthogonal projection of $C$1 onto the approximate normal space. The geometric mechanism is local flattening. Near a rectifiable point $C$2, the set looks like its tangent $C$3-plane, so $C$4 is asymptotically governed only by the part of $C$5 visible in the normal directions. Tangential components of $C$6 do not contribute to the leading $C$7-term (Fryš, 30 Jan 2026).

The same paper extends the formula to countably $C$8-rectifiable compact sets under an AFP-type hypothesis. In the full-dimensional case $C$9, the relevant condition is the AFP-$C$0-condition. For lower-dimensional anisotropy, when $C$1 and $C$2, the generalized condition is the AFP-$C$3-condition relative to $C$4: $C$5 for some $C$6 and some Radon measure $C$7. If $C$8, this again yields

$C$9

for every $C\in\mathcal C_0^n$0 (Fryš, 30 Jan 2026).

Two threshold cases are completely described. If $C\in\mathcal C_0^n$1, no AFP assumption is needed and there is a slicing formula

$C\in\mathcal C_0^n$2

which reduces to the same tangent-plane formula for countably rectifiable $C\in\mathcal C_0^n$3. If $C\in\mathcal C_0^n$4, then

$C\in\mathcal C_0^n$5

because the projected anisotropy has dimension $C\in\mathcal C_0^n$6, so $C\in\mathcal C_0^n$7 almost everywhere (Fryš, 30 Jan 2026).

The codimension-one specialization recovers the familiar support-function density. When $C\in\mathcal C_0^n$8, the normal space is one-dimensional and

$C\in\mathcal C_0^n$9

hence

$S\subset \mathbb R^n$00

For $S\subset \mathbb R^n$01, the projected normal section is the Euclidean unit ball in the normal space, so $S\subset \mathbb R^n$02, and the formula reduces to Federer’s isotropic identification $S\subset \mathbb R^n$03 (Fryš, 30 Jan 2026).

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 $S\subset \mathbb R^n$04 be compact and countably $S\subset \mathbb R^n$05-rectifiable, and let $S\subset \mathbb R^n$06 be a nonempty compact convex set with $S\subset \mathbb R^n$07. The key sufficient hypothesis is the AFP-condition relative to $S\subset \mathbb R^n$08: $S\subset \mathbb R^n$09 for some Radon measure $S\subset \mathbb R^n$10 and $S\subset \mathbb R^n$11. Under this weaker, directionally adapted condition, the $S\subset \mathbb R^n$12-Minkowski content exists and satisfies

$S\subset \mathbb R^n$13

where $S\subset \mathbb R^n$14 is a unit normal defined up to sign. The symmetral $S\subset \mathbb R^n$15 appears because $S\subset \mathbb R^n$16 expands on both sides of the hypersurface, so only the symmetric width matters (Kiderlen et al., 4 Apr 2025).

This relative AFP condition is strictly weaker than the classical Ambrosio–Fusco–Pallara density condition when $S\subset \mathbb R^n$17. The right-hand side measures only the transverse $S\subset \mathbb R^n$18-dimensional spread of $S\subset \mathbb R^n$19 through $S\subset \mathbb R^n$20, together with the expected $S\subset \mathbb R^n$21 behavior along the $S\subset \mathbb R^n$22-directions. Accordingly, lower-dimensional thickening can succeed even when isotropic thickening fails (Kiderlen et al., 4 Apr 2025).

The corresponding outer theory identifies one-sided growth with anisotropic perimeter. For a set $S\subset \mathbb R^n$23 of finite perimeter,

$S\subset \mathbb R^n$24

and if $S\subset \mathbb R^n$25 admits the $S\subset \mathbb R^n$26-Minkowski content with

$S\subset \mathbb R^n$27

then $S\subset \mathbb R^n$28 admits outer $S\subset \mathbb R^n$29-Minkowski content and

$S\subset \mathbb R^n$30

A practical criterion is obtained when $S\subset \mathbb R^n$31 is compact, satisfies the AFP-condition relative to $S\subset \mathbb R^n$32, and

$S\subset \mathbb R^n$33

then $S\subset \mathbb R^n$34 (Kiderlen et al., 4 Apr 2025).

A particularly sharp one-dimensional directed case is $S\subset \mathbb R^n$35, $S\subset \mathbb R^n$36. Then

$S\subset \mathbb R^n$37

and the result holds for every finite-perimeter set after choosing a suitable representative of its $S\subset \mathbb R^n$38-class. This is exceptional in that no AFP-type assumption is required (Kiderlen et al., 4 Apr 2025).

The geometric distinction from isotropic thickening is visible in explicit examples. One example constructs a compact finite-perimeter set $S\subset \mathbb R^n$39 such that isotropic outer Minkowski content does not exist, while $S\subset \mathbb R^n$40 exists for every two-dimensional unit disk $S\subset \mathbb R^n$41. Another product example gives $S\subset \mathbb R^n$42 with pathological isotropic outer growth in the complementary factor, yet existence of outer content for $S\subset \mathbb R^n$43. 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 (Kiderlen et al., 4 Apr 2025).

5. Anisotropic $S\subset \mathbb R^n$44-content, Kneser functions, and dimensions

A further development compares anisotropic lower-dimensional Minkowski content with anisotropic $S\subset \mathbb R^n$45-content, the latter being defined through anisotropic perimeter of parallel sets. For compact $S\subset \mathbb R^n$46,

$S\subset \mathbb R^n$47

and away from an at most countable exceptional set,

$S\subset \mathbb R^n$48

The anisotropic $S\subset \mathbb R^n$49-content is then

$S\subset \mathbb R^n$50

with the usual convention when lower and upper values agree (Fryš, 8 Sep 2025).

The structural theorem behind the comparison is that

$S\subset \mathbb R^n$51

is a Kneser function of order $S\subset \mathbb R^n$52 for every compact $S\subset \mathbb R^n$53 and every $S\subset \mathbb R^n$54. This extends the classical Euclidean result and the symmetric-convex case to arbitrary convex bodies with $S\subset \mathbb R^n$55. The proof uses the monotonicity of

$S\subset \mathbb R^n$56

which is decreasing in $S\subset \mathbb R^n$57 (Fryš, 8 Sep 2025).

For $S\subset \mathbb R^n$58, the Kneser framework yields the basic anisotropic inequalities

$S\subset \mathbb R^n$59

There is also a sharper upper comparison,

$S\subset \mathbb R^n$60

and a lower isoperimetric comparison with shifted dimension parameter,

$S\subset \mathbb R^n$61

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 (Fryš, 8 Sep 2025).

The corresponding dimensions reflect the same pattern. The upper anisotropic Minkowski and $S\subset \mathbb R^n$62-dimensions coincide,

$S\subset \mathbb R^n$63

and the upper $S\subset \mathbb R^n$64-dimension is independent of $S\subset \mathbb R^n$65. For lower dimensions one has only

$S\subset \mathbb R^n$66

This mirrors the earlier isotropic theory, where explicit examples show that lower surface-based and lower volume-based dimensions can separate sharply (Winter, 2010, Fryš, 8 Sep 2025).

The planar Sierpiński gasket gives a concrete anisotropic example. For $S\subset \mathbb R^n$67, the $S\subset \mathbb R^n$68-dimensional $S\subset \mathbb R^n$69-anisotropic Minkowski content and anisotropic $S\subset \mathbb R^n$70-content do not exist for any $S\subset \mathbb R^n$71; rather,

$S\subset \mathbb R^n$72

Here the anisotropy enters through the support-function sum

$S\subset \mathbb R^n$73

where $S\subset \mathbb R^n$74 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 (Fryš, 8 Sep 2025).

6. Fractal curvature-direction measures, dependence on $S\subset \mathbb R^n$75, and scope

For self-conformal sets, the lower-dimensional anisotropic picture becomes measure-valued and typically averaged. If $S\subset \mathbb R^n$76 is self-conformal with dimension $S\subset \mathbb R^n$77, the $S\subset \mathbb R^n$78-th fractal curvature-direction measure is defined, in Cesàro form, by

$S\subset \mathbb R^n$79

as a measure on $S\subset \mathbb R^n$80. For $S\subset \mathbb R^n$81 this is averaged Minkowski content, for $S\subset \mathbb R^n$82 averaged surface content, and for $S\subset \mathbb R^n$83 it yields lower-order, lower-dimensional contents extracted from parallel sets. Because the measures live on $S\subset \mathbb R^n$84, they are genuinely directional and hence anisotropic (Bohl, 2012).

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 $S\subset \mathbb R^n$85, 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 $S\subset \mathbb R^n$86, nontrivial uniform integrability conditions controlling curvature on overlap regions (Bohl, 2012).

Dependence on the anisotropy also bifurcates according to whether $S\subset \mathbb R^n$87 is full-dimensional. For countably $S\subset \mathbb R^n$88-rectifiable compact sets and full-dimensional convex bodies $S\subset \mathbb R^n$89, existence with the geometric value for one $S\subset \mathbb R^n$90 is equivalent to existence with the corresponding value for every other full-dimensional $S\subset \mathbb R^n$91. By contrast, when $S\subset \mathbb R^n$92 is lower-dimensional, dependence on $S\subset \mathbb R^n$93 can be genuine because the span $S\subset \mathbb R^n$94 selects the directions along which the set is probed, and the relevant AFP condition must be formulated relative to $S\subset \mathbb R^n$95 (Fryš, 30 Jan 2026).

Several limitations remain intrinsic to the subject. Codimension-one results such as the outer-content theorem for finite-perimeter sets do not address $S\subset \mathbb R^n$96 directly (Chambolle et al., 2012). The transfer principle for boundary Minkowski content is also codimension-one (Fryš, 11 Aug 2025). The theory with lower-dimensional structuring elements provides strong sufficient conditions, not a complete characterization of when

$S\subset \mathbb R^n$97

holds in full generality (Kiderlen et al., 4 Apr 2025). The fractal self-conformal theory is predominantly an averaged theory rather than an ordinary pointwise one (Bohl, 2012). 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 $S\subset \mathbb R^n$98 or their symmetrized version. In arbitrary codimension, the correct density is the $S\subset \mathbb R^n$99-dimensional volume of the projection of $k$00 onto the approximate normal space,

$k$01

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 (Fryš, 30 Jan 2026, Bohl, 2012).