---
title: S-Measure and Minkowski Equivalence
url: https://www.emergentmind.com/topics/s-measure
type: topic
---

# S-Measure and Minkowski Equivalence

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S-measure, in the sense developed through the surface-area based content of parallel sets, is the notion of measuring a bounded set \(A \subset \mathbb{R}^d\) through the asymptotic behavior of the surface area of the boundary of its \(r\)-parallel neighborhood as \(r \to 0\). In the formulation of Rataj and Winter, the relevant object is the \(s\)-dimensional S-content, defined in direct analogy with the classical Minkowski content, and a set is called \(s\)-S-measurable when this content exists and is positive and finite. The central result is that Minkowski measurability and S-measurability coincide, with exact agreement of the corresponding limits, and that this equivalence extends to two-sided positivity and finiteness bounds as well as to generalized contents defined by gauge functions [1111.1825].

## 1. Parallel sets and the classical Minkowski content

Let \(A \subset \mathbb{R}^d\) be bounded. The Euclidean distance from \(x\) to \(A\) is
\[
d_A(x)=\inf_{a\in A}|x-a|.
\]
For \(r>0\), the \(r\)-parallel neighborhood is
\[
A_r = \{\,x\in\mathbb{R}^d : d_A(x) \le r\,\}.
\]
Its \(d\)-dimensional volume is
\[
V_r(A) := \operatorname{vol}_d(A_r)=\mathcal{H}^d(A_r).
\]

Fix \(s \in [0,d]\). The \(s\)-dimensional lower and upper Minkowski contents of \(A\) are defined by
\[
\underline M_s(A):=\liminf_{r\to0}\frac{V_r(A)}{r^{\,d-s}\,\kappa_{d-s}},
\qquad
\overline M_s(A):=\limsup_{r\to0}\frac{V_r(A)}{r^{\,d-s}\,\kappa_{d-s}},
\]
where
\[
\kappa_t=\frac{\pi^{t/2}}{\Gamma(1+t/2)}
\]
is the volume of the unit \(t\)-ball. Whenever
\[
\underline M_s(A)=\overline M_s(A)\in(0,\infty),
\]
their common value \(M_s(A)\) is called the \(s\)-dimensional Minkowski content of \(A\), and \(A\) is said to be \(s\)-Minkowski measurable [1111.1825].

This framework measures the small-scale growth of the volume of parallel sets. In the terminology used by Rataj and Winter, the S-content is introduced in complete analogy with this classical volume-based construction.

## 2. Definition of S-content and S-measurability

The surface-area analogue is obtained by setting
\[
S_r(A):=\mathcal{H}^{\,d-1}(\partial A_r),
\]
the surface area of the boundary of \(A_r\). The lower and upper \(s\)-dimensional S-contents are then defined by
\[
\underline S_s(A):=\liminf_{r\to0}\frac{S_r(A)}{r^{\,d-1-s}\,(d-s)\,\kappa_{d-s}},
\qquad
\overline S_s(A):=\limsup_{r\to0}\frac{S_r(A)}{r^{\,d-1-s}\,(d-s)\,\kappa_{d-s}}.
\]
If
\[
\underline S_s(A)=\overline S_s(A)\in(0,\infty),
\]
their common value \(S_s(A)\) is called the \(s\)-dimensional S-content of \(A\), and \(A\) is said to be \(s\)-S-measurable [1111.1825].

The normalization is chosen so that the asymptotic scaling of \(S_r(A)\) matches that of the derivative of \(V_r(A)\). This suggests that S-content is not merely analogous to Minkowski content but is structurally linked to it through the geometry of parallel sets.

## 3. Equivalence with Minkowski measurability

The central theorem states that Minkowski measurability and S-measurability coincide. Let \(A \subset \mathbb{R}^d\) be bounded, let \(D \in [0,d)\), and set \(s=d-D\). Then the following are equivalent:

1. \(A\) is \(D\)-Minkowski measurable, that is,
   \[
   \lim_{r\to0}\frac{V_r(A)}{r^s\,\kappa_s}=M
   \quad\text{with } 0<M<\infty.
   \]

2. \(A\) is \(D\)-S-measurable, that is,
   \[
   \lim_{r\to0}\frac{S_r(A)}{r^{s-1}\,(d-s)\,\kappa_s}=S
   \quad\text{with } 0<S<\infty.
   \]

Moreover, in that case one has the exact coincidence of the two limits:
\[
\lim_{r\to0}\frac{V_r(A)}{r^s\,\kappa_s}
=
\lim_{r\to0}\frac{S_r(A)}{r^{s-1}\,(d-s)\,\kappa_s}.
\]
In particular, the common exponent \(D\) is then both the Minkowski and the S-dimension of \(A\) [1111.1825].

A further theorem gives the corresponding two-sided criterion without requiring existence of a limit. For bounded \(A \subset \mathbb{R}^d\) and fixed \(D \in [0,d)\),
\[
0<\underline M_D(A)\le\overline M_D(A)<\infty
\]
if and only if
\[
0<\underline S_D(A)\le\overline S_D(A)<\infty.
\]
Thus any two-sided positive finite bound for the Minkowski content at exponent \(D\) forces the same two-sided bound for the S-content, and vice versa [1111.1825].

These results characterize S-measure as a fully equivalent surface-area formulation of Minkowski measurability rather than a weaker proxy. A plausible implication is that, for bounded sets in Euclidean space, the asymptotic geometry of parallel volume and parallel surface area carries the same measurable information at the critical exponent.

## 4. Kneser functions and the analytic mechanism

The equivalence is derived from analogous statements for Kneser functions. A function \(f:(0,\infty)\to(0,\infty)\) is a Kneser function of order \(d>1\) if for all \(0<a<b\) and \(\lambda\ge1\) one has
\[
f(\lambda b)-f(\lambda a)\le \lambda^d [\,f(b)-f(a)\,].
\]

Rataj and Winter establish two general principles for such functions. First, if \(f\) is Kneser of order \(d\) and
\[
0<\liminf_{r\to0}\frac{f(r)}{r^{d-s}}
\le
\limsup_{r\to0}\frac{f(r)}{r^{d-s}}
<\infty,
\]
then also
\[
0<\liminf_{r\to0}\frac{f'(r)}{r^{d-s-1}}
\le
\limsup_{r\to0}\frac{f'(r)}{r^{d-s-1}}
<\infty,
\]
and conversely. Second, if in addition the limit
\[
\lim_{r\to0}\frac{f(r)}{r^{d-s}}=M\in(0,\infty)
\]
exists, then
\[
\lim_{r\to0}\frac{f'(r)}{r^{d-s-1}}=M
\]
also exists and equals \(M\) [1111.1825].

These principles are applied to the volume function \(f(r)=V_r(A)\), whose left and right derivatives satisfy, in the sense of measures, \(f'(r)\sim S_r(A)\). This immediately yields the corresponding statements for Minkowski and S-contents. The role of the Kneser-function framework is therefore foundational: it transfers asymptotic information between a function and its derivative, and in the geometric setting those two objects are the volume and surface area of parallel sets.

## 5. Generalized contents and gauge functions

The theory extends beyond power-law normalizations. Let \(h:(0,\infty)\to(0,\infty)\) be a continuous gauge function with \(h(r)\to0\) as \(r\to0\). The generalized Minkowski contents are defined by
\[
\underline M(h;A)=\liminf_{r\to0}\frac{V_r(A)}{h(r)},
\qquad
\overline M(h;A)=\limsup_{r\to0}\frac{V_r(A)}{h(r)},
\]
and similarly for the generalized S-contents with the same \(h\).

Under mild regularity assumptions on \(h\), one again has an equivalence theorem: if \(h\) is differentiable near \(0\), \(h'\) does not vanish, and the generalized Minkowski content \(M(h;A)\) exists in \((0,\infty)\), then the generalized S-content \(S(h';A)\) exists in \((0,\infty)\) and equals \(M(h;A)\), and conversely [1111.1825].

In particular, for
\[
h(r)=r^{d-D}g(r)
\]
with \(g\) nondecreasing and slowly varying, one has
\[
M(h;A)=M \;\Rightarrow\; S(h';A)=M,
\]
and conversely. This shows that the relation between Minkowski content and S-content is not restricted to pure dimensional scaling. It persists for more general gauges, provided the gauge has sufficient differentiability and nondegeneracy near the origin.

## 6. One-dimensional fractal strings and the Modified Weyl–Berry conjecture

In dimension \(d=1\), the theory specializes to compact sets \(F\subset\mathbb{R}\) of Minkowski dimension \(D\in(0,1)\), studied via their complementary intervals, or fractal string, \(\mathcal{L}=(\ell_j)_j\). Lapidus–Pomerance showed that \(F\) is Minkowski measurable of dimension \(D\) if and only if
\[
\ell_j \approx j^{-1/D},
\]
and used this to resolve the modified Weyl–Berry conjecture for planar vibrations [1111.1825].

Rataj and Winter add a further equivalent criterion. For compact \(F\subset\mathbb{R}\) with \(\dim_M F=D\in(0,1)\), the following are all equivalent:
- \(0<\underline M_D(F)\le\overline M_D(F)<\infty\),
- \(0<\underline S_D(F)\le\overline S_D(F)<\infty\),
- \(\ell_j \sim j^{-1/D}\) as \(j\to\infty\).

Moreover, exact measurability satisfies
\[
\lim \frac{V_r(F)}{r^{1-D}}=M>0
\quad\Longleftrightarrow\quad
\lim \frac{S_r(F)}{r^{-D}}=M>0
\quad\Longleftrightarrow\quad
\ell_j \sim L\cdot j^{-1/D},
\]
and in this case
\[
M_D(F)=S_D(F)=2^{1-D}\,\frac{L}{1-D}\,\kappa_{1-D}.
\]
By passing through the S-content, several steps in the original Lapidus–Pomerance proofs can be streamlined, in particular the sharp two-sided estimates of the counting error for eigenvalues of the Dirichlet Laplacian on the complement of \(F\) [1111.1825].

This one-dimensional application situates S-measure within spectral asymptotics and fractal-string theory. It also shows that the surface-area formulation is not only equivalent at an abstract level but can serve as an effective intermediate criterion in problems where interval asymptotics and geometric content are linked.

Source: https://www.emergentmind.com/topics/s-measure