Anisotropic Minkowski Content

• Anisotropic Minkowski content is defined by replacing Euclidean balls with a convex body in tubular neighborhood constructions, capturing first-order volume growth as anisotropic perimeter.
• It extends to lower-dimensional settings by studying r^(n–s) asymptotics, where projections of the convex body onto normal spaces determine rectifiable set measures.
• It underpins anisotropic minimal hypersurface theory by linking normalized energy and density, enabling monotonicity formulas and Γ‐convergence results in geometric analysis.

Searching arXiv for recent and foundational papers on anisotropic Minkowski content, anisotropic outer Minkowski content, and related anisotropic density formulations. Anisotropic Minkowski content is a family of asymptotic geometric quantities obtained by replacing Euclidean balls in tubular-neighborhood constructions with a convex body or, equivalently in many settings, with the unit ball of a Minkowski norm. In the codimension-one finite-perimeter setting, it measures the first-order volume growth of anisotropic dilations and identifies that growth with anisotropic perimeter. In lower-dimensional settings, it studies the asymptotics of $\lambda^n(E\oplus rC)$ at scale $r^{n-s}$. In anisotropic minimal hypersurface theory, a closely related density is furnished by the normalized anisotropic energy $r^{-n}\int_{M\cap(r\Omega)}F(\nu)$, which plays the role of an anisotropic Minkowski content even though that terminology is not introduced explicitly in that context (Chambolle et al., 2012, Fryš, 30 Jan 2026, Pham, 13 Jan 2026).

1. Core definitions and geometric framework

The basic anisotropic ingredient is a convex body $C\subset \mathbb R^n$ with $0$ in its interior, or a Minkowski norm $F$. For a convex body $C$, the support function is

$h_C(\nu)=\sup_{x\in C} x\cdot \nu,$

and the corresponding polar gauge is

$$h_C^\circ(x)=\sup_{h_C(\nu)\le 1} x\cdot \nu, \qquad C=\{h_C^\circ\le 1\}.$$

The anisotropic tubular neighborhood is defined by Minkowski addition,

$E_{r,C}:=E\oplus rC=\{e+rc:\ e\in E,\ c\in C\},$

and the associated anisotropic volume function is

$r^{n-s}$0

These constructions are the common substrate for outer content, lower-dimensional content, and anisotropic density theory (Chambolle et al., 2012, Fryš, 8 Sep 2025).

Setting	Representative formula	Limit object
Outer content	$r^{n-s}$1	anisotropic perimeter
Lower-dimensional content	$r^{n-s}$2	$r^{n-s}$3-dimensional anisotropic content
Boundary content	$r^{n-s}$4	boundary measure
Minimal-hypersurface density	$r^{n-s}$5	anisotropic density

Several closely related normalizations are used. One line of work defines

$r^{n-s}$6

while another writes

$r^{n-s}$7

This indicates that the underlying geometry is the same, but the normalization convention can differ across papers (Fryš, 8 Sep 2025, Fryš, 30 Jan 2026).

In codimension one, anisotropic perimeter is the principal limit object. For a finite-perimeter set $r^{n-s}$8,

$r^{n-s}$9

and equivalently

$r^{-n}\int_{M\cap(r\Omega)}F(\nu)$0

in the measure-theoretic sense. When $r^{-n}\int_{M\cap(r\Omega)}F(\nu)$1, $r^{-n}\int_{M\cap(r\Omega)}F(\nu)$2, so the anisotropic theory reduces to the classical isotropic one (Chambolle et al., 2012).

2. Outer anisotropic Minkowski content and the perimeter correspondence

The classical anisotropic outer-content problem asks for the existence of the limit

$r^{-n}\int_{M\cap(r\Omega)}F(\nu)$3

or, in localized form,

$r^{-n}\int_{M\cap(r\Omega)}F(\nu)$4

The central codimension-one theorem states that, on the same class of finite-perimeter sets for which the classical outer Minkowski content exists and equals perimeter, the anisotropic outer Minkowski content exists and equals the anisotropic perimeter (Chambolle et al., 2012).

More precisely, if

$r^{-n}\int_{M\cap(r\Omega)}F(\nu)$5

then

$r^{-n}\int_{M\cap(r\Omega)}F(\nu)$6

Thus the first-order growth of the anisotropic dilation $r^{-n}\int_{M\cap(r\Omega)}F(\nu)$7 is governed by

$r^{-n}\int_{M\cap(r\Omega)}F(\nu)$8

In the convex/smooth case, the familiar expansion

$r^{-n}\int_{M\cap(r\Omega)}F(\nu)$9

is recovered as a special case, and the theorem extends that picture to a much broader finite-perimeter class (Chambolle et al., 2012).

The same paper establishes a variational counterpart: both the outer-content functional and its associated set functional $C\subset \mathbb R^n$0-converge in $C\subset \mathbb R^n$1 to the anisotropic perimeter. In functional form, the full family $C\subset \mathbb R^n$2 $C\subset \mathbb R^n$3-converges to the anisotropic total variation

$C\subset \mathbb R^n$4

This places anisotropic Minkowski content within the standard $C\subset \mathbb R^n$5/$C\subset \mathbb R^n$6-convergence framework (Chambolle et al., 2012).

A later equivalence theorem sharpens the codimension-one picture. For a finite-perimeter set $C\subset \mathbb R^n$7 and $C\subset \mathbb R^n$8,

$C\subset \mathbb R^n$9

if and only if the analogous identity holds for $0$0. Hence the property that the boundary Minkowski content agrees with the expected anisotropic perimeter average is independent of the choice of convex body (Fryš, 11 Aug 2025).

This equivalence has a strong consequence for outer content. If $0$1 and, for some $0$2,

$0$3

then for every $0$4,

$0$5

In particular, anisotropic outer-content existence for a set and its complement implies the isotropic outer Minkowski content for both (Fryš, 11 Aug 2025).

3. Lower-dimensional anisotropic content and rectifiable sets

The lower-dimensional theory replaces the codimension-one perimeter paradigm by the asymptotics of $0$6 at order $0$7. For compact $0$8 and $0$9, one defines

$F$0

and, when the two agree, their common value $F$1 (Fryš, 8 Sep 2025).

A central rectifiable-set result states that the $F$2-anisotropic $F$3-dimensional Minkowski content of a $F$4-rectifiable compact set always exists and is determined by the geometry of the projections of $F$5 onto the normal spaces. For a countably $F$6-rectifiable set $F$7, at $F$8-a.e. $F$9 one considers the approximate tangent plane $C$0, the normal space

$C$1

and the projected convex body

$C$2

The paper states that the limiting functional depends on $C$3, making the anisotropy explicitly orientation dependent (Fryš, 30 Jan 2026).

This dependence admits a radial description. The paper records

$C$4

so the local density is controlled by the $C$5-dimensional volume of the projection of $C$6 onto the normal slice. In the isotropic case $C$7, the projection is always the unit ball in the normal space, and the orientation dependence disappears (Fryš, 30 Jan 2026).

For countably $C$8-rectifiable compact sets, existence requires an AFP-type density condition. One formulation is: there exist $C$9 and a Radon measure $h_C(\nu)=\sup_{x\in C} x\cdot \nu,$0 such that

$h_C(\nu)=\sup_{x\in C} x\cdot \nu,$1

Under this hypothesis, the same limiting formula extends from rectifiable compact sets to countably rectifiable compact sets (Fryš, 30 Jan 2026).

A related paper treats the case where the structuring element $h_C(\nu)=\sup_{x\in C} x\cdot \nu,$2 may be lower-dimensional. For a compact convex set $h_C(\nu)=\sup_{x\in C} x\cdot \nu,$3 contained in a $h_C(\nu)=\sup_{x\in C} x\cdot \nu,$4-dimensional subspace $h_C(\nu)=\sup_{x\in C} x\cdot \nu,$5, a relative AFP-condition adapted to $h_C(\nu)=\sup_{x\in C} x\cdot \nu,$6 is introduced: $h_C(\nu)=\sup_{x\in C} x\cdot \nu,$7 This is weaker than the usual full-dimensional AFP-condition when $h_C(\nu)=\sup_{x\in C} x\cdot \nu,$8, and suffices for the existence of $h_C(\nu)=\sup_{x\in C} x\cdot \nu,$9-Minkowski content in the lower-dimensional setting (Kiderlen et al., 4 Apr 2025).

The lower-dimensional theory also shows that anisotropic content can exist when the isotropic one does not. In a three-dimensional example,

$$h_C^\circ(x)=\sup_{h_C(\nu)\le 1} x\cdot \nu, \qquad C=\{h_C^\circ\le 1\}.$$0

so isotropic outer Minkowski content fails, yet the outer $$h_C^\circ(x)=\sup_{h_C(\nu)\le 1} x\cdot \nu, \qquad C=\{h_C^\circ\le 1\}.$$1-Minkowski content exists for every two-dimensional disk $$h_C^\circ(x)=\sup_{h_C(\nu)\le 1} x\cdot \nu, \qquad C=\{h_C^\circ\le 1\}.$$2. This demonstrates that lower-dimensional anisotropic thickening may admit a clean first-order asymptotic even when full-dimensional Euclidean thickening does not (Kiderlen et al., 4 Apr 2025).

4. Anisotropic $$h_C^\circ(x)=\sup_{h_C(\nu)\le 1} x\cdot \nu, \qquad C=\{h_C^\circ\le 1\}.$$3-content, Kneser functions, and dimension theory

A substantial extension of the subject introduces anisotropic $$h_C^\circ(x)=\sup_{h_C(\nu)\le 1} x\cdot \nu, \qquad C=\{h_C^\circ\le 1\}.$$4-content through the derivative of the anisotropic volume function. For compact $$h_C^\circ(x)=\sup_{h_C(\nu)\le 1} x\cdot \nu, \qquad C=\{h_C^\circ\le 1\}.$$5 and $$h_C^\circ(x)=\sup_{h_C(\nu)\le 1} x\cdot \nu, \qquad C=\{h_C^\circ\le 1\}.$$6,

$$h_C^\circ(x)=\sup_{h_C(\nu)\le 1} x\cdot \nu, \qquad C=\{h_C^\circ\le 1\}.$$7

Since

$$h_C^\circ(x)=\sup_{h_C(\nu)\le 1} x\cdot \nu, \qquad C=\{h_C^\circ\le 1\}.$$8

one defines lower and upper anisotropic $$h_C^\circ(x)=\sup_{h_C(\nu)\le 1} x\cdot \nu, \qquad C=\{h_C^\circ\le 1\}.$$9-contents by

$E_{r,C}:=E\oplus rC=\{e+rc:\ e\in E,\ c\in C\},$0

This construction is derivative-based, in contrast with the volume-based definition of $E_{r,C}:=E\oplus rC=\{e+rc:\ e\in E,\ c\in C\},$1 (Fryš, 8 Sep 2025).

The key analytic input is that the anisotropic volume function is of Kneser type of order $E_{r,C}:=E\oplus rC=\{e+rc:\ e\in E,\ c\in C\},$2: $E_{r,C}:=E\oplus rC=\{e+rc:\ e\in E,\ c\in C\},$3 This yields local absolute continuity, one-sided derivatives everywhere, and comparison inequalities between asymptotics of $E_{r,C}:=E\oplus rC=\{e+rc:\ e\in E,\ c\in C\},$4 and $E_{r,C}:=E\oplus rC=\{e+rc:\ e\in E,\ c\in C\},$5 (Fryš, 8 Sep 2025).

Applied with $E_{r,C}:=E\oplus rC=\{e+rc:\ e\in E,\ c\in C\},$6, the Kneser framework gives

$E_{r,C}:=E\oplus rC=\{e+rc:\ e\in E,\ c\in C\},$7

When $E_{r,C}:=E\oplus rC=\{e+rc:\ e\in E,\ c\in C\},$8, $E_{r,C}:=E\oplus rC=\{e+rc:\ e\in E,\ c\in C\},$9 and $r^{n-s}$00 coincide, so one obtains

$r^{n-s}$01

Thus anisotropic Minkowski content and anisotropic $r^{n-s}$02-content are comparable but need not agree (Fryš, 8 Sep 2025).

The same paper develops anisotropic Minkowski and $r^{n-s}$03-dimensions. It proves that the anisotropic Minkowski dimension does not depend on $r^{n-s}$04, and that the upper anisotropic $r^{n-s}$05-dimension is also independent of $r^{n-s}$06. Moreover, for $r^{n-s}$07,

$r^{n-s}$08

which implies equality of the upper dimensions: $r^{n-s}$09 By contrast, the lower dimensions need not coincide (Fryš, 8 Sep 2025).

The Sierpiński gasket supplies an explicit anisotropic counterexample to naive existence expectations. At dimension $r^{n-s}$10, the paper finds

$r^{n-s}$11

Hence neither the anisotropic Minkowski content nor the anisotropic $r^{n-s}$12-content exists at that dimension, even for a classical self-similar fractal (Fryš, 8 Sep 2025).

5. Density formulations for anisotropic minimal hypersurfaces

A different but closely related use of anisotropic Minkowski-type asymptotics appears in anisotropic minimal hypersurface theory. Let

$r^{n-s}$13

be a smooth, positively $r^{n-s}$14-homogeneous Minkowski norm with Wulff shape $r^{n-s}$15, where $r^{n-s}$16 is a bounded uniformly convex domain containing the origin, and let $r^{n-s}$17 be the dual Minkowski norm. For an oriented smooth hypersurface $r^{n-s}$18 with unit normal $r^{n-s}$19, the anisotropic normal and anisotropic mean curvature are

$r^{n-s}$20

The hypersurface is $r^{n-s}$21-minimal when $r^{n-s}$22 (Pham, 13 Jan 2026).

The central monotonicity formula states that if $r^{n-s}$23 is an oriented smooth $r^{n-s}$24-minimal hypersurface and $r^{n-s}$25, then for every $r^{n-s}$26,

$r^{n-s}$27

Under the sign condition

$r^{n-s}$28

the quantity

$r^{n-s}$29

is monotone increasing, and it is constant if and only if $r^{n-s}$30 is a hyperplane (Pham, 13 Jan 2026).

This paper does not explicitly define an object called anisotropic Minkowski content, but it identifies the normalized anisotropic energy

$r^{n-s}$31

as the anisotropic analogue of the Euclidean normalized area ratio. In that sense, $r^{n-s}$32 functions as an anisotropic Minkowski content or density. The small-scale limit exists for smooth $r^{n-s}$33 through the origin and is computed as

$r^{n-s}$34

The corresponding sharp lower bound is

$r^{n-s}$35

with equality if and only if $r^{n-s}$36 is a hyperplane (Pham, 13 Jan 2026).

The isotropic case $r^{n-s}$37 recovers the classical monotonicity formula

$r^{n-s}$38

A simple anisotropic class satisfying the sign condition is

$r^{n-s}$39

for a positive definite symmetric matrix $r^{n-s}$40, with dual norm

$r^{n-s}$41

and identity

$r^{n-s}$42

This shows that anisotropic density theory can be formulated directly from the ambient Minkowski norm and its dual (Pham, 13 Jan 2026).

6. Related notions, special cases, and conceptual boundaries

The phrase “anisotropic Minkowski” is used in several neighboring but distinct senses. One should distinguish anisotropic Minkowski content from the anisotropic Minkowski problem, which prescribes anisotropic Gauss–Kronecker curvature for closed strongly convex hypersurfaces via a Monge–Ampère equation on the anisotropic support function (Xia, 2012). It is likewise distinct from anisotropic Minkowski inequalities, where one studies inequalities such as

$r^{n-s}$43

or the $r^{n-s}$44-capacity-based inequalities for anisotropic mean curvature integrals (Xia et al., 2020). These are Minkowski-type in the convex-geometric sense, but they are not definitions of content.

A second source of potential confusion is the tensorial literature. Minkowski tensors quantify anisotropy of morphology through tensor-valued valuations such as

$r^{n-s}$45

or, in random-field form, through interfacial tensor densities $r^{n-s}$46. These are anisotropy-sensitive refinements of scalar Minkowski functionals, but they are not tubular-growth contents. Their role is to measure directional organization, not first-order neighborhood volume asymptotics (Schröder-Turk et al., 2010, Klatt et al., 2021).

Across the content literature, several special cases recur. If $r^{n-s}$47, the anisotropic constructions recover the classical isotropic Minkowski content and perimeter. If $r^{n-s}$48 is lower-dimensional, the first-order asymptotics are governed by the support function of $r^{n-s}$49, or equivalently by the convex hull of $r^{n-s}$50 with the origin in the outer-content formula. For $r^{n-s}$51, the lower-dimensional rectifiable-set theory reduces to the surface-tension expression

$r^{n-s}$52

which matches the codimension-one boundary-content paradigm (Kiderlen et al., 4 Apr 2025, Fryš, 30 Jan 2026).

The modern theory therefore presents anisotropic Minkowski content not as a single invariant but as a coherent family of asymptotic boundary and tubular-growth quantities. In codimension one, it identifies anisotropic perimeter as the first-order growth coefficient of $r^{n-s}$53-dilation. In lower dimensions, it depends on the interaction between the tangent geometry of the set and the projections of $r^{n-s}$54 onto normal spaces. In fractal and derivative-based settings, it interfaces with anisotropic $r^{n-s}$55-content and dimension theory. In anisotropic minimal hypersurface theory, an energy-normalized monotonicity quantity plays the same structural role as a density. This suggests a unified interpretation: anisotropic Minkowski content records how the geometry of a chosen convex body or Minkowski norm weights infinitesimal thickening, and therefore how anisotropy enters the passage from volume growth to boundary measure.