Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 27, 2002, 437–444

WALL PROPERTIES OF DOMAINS IN EUCLIDEAN SPACES

Jussi Väisälä

Helsingin yliopisto, Matematiikan laitos PL 4 (Yliopistonkatu 5), FI-00014 Helsinki, Finland; jvaisala@cc.helsinki.fi

Abstract. Wall theorems give local lower bounds for the p-measure of the boundary of a domain in the euclidean n-space. We improve earlier results by replacing the euclidean metric by the inner metric of the domain and also the Hausdorff p-content by the projectional p-measure.

1. Introduction

Wall properties of domains in the euclidean n-space R^{n} were first considered by J. Heinonen [He] in 1996 in connection with quasiconformal maps. Suppose that G ⊂ R^{n} is a domain, K -quasiconformally equivalent to a ball. Let a ∈ G and set r = d(a, ∂G). Heinonen made the following quasiconformal wall conjecture:

(1.1)  m_{n−1}(∂G ∩ B(a, 2r)) ≥ r^{n−1}/c,

where m_{n−1} is the (n − 1)-dimensional Hausdorff measure, and the constant c depends only on K and n. The case n = 2 is trivial, and the case n = 3 was proved in [He, Section 5].

The author [Vä] proved a result that implied the wall conjecture in all dimensions. Instead of quasiconformality, the result was formulated in terms of homological [Vä, 1.3] or homotopical [Vä, 6.2] connectivity properties of the domain G. Moreover, the result gave a lower bound for the Hausdorff p-measure m_{p}, 1 ≤ p ≤ n − 1, and in fact, for the Hausdorff p-content m_{p}^{∞}. We recall the homotopical version.

1.2. Theorem ([Vä, 6.2]). Let n ≥ 2 and 1 ≤ p ≤ n − 1 be integers and let c ≥ 1. Suppose that G ⊊ R^{n} is a domain with the following properties:

(1) Every map f: S^{n−1} → G is null-homotopic.

(2) If x ∈ R^{n}, t > 0, n − p ≤ k ≤ n − 2, then every map f: S^{k} → G ∩ B(x, t) is null-homotopic in G ∩ B(x, ct).

Let a ∈ G and set r = d(a, ∂G). Then

m_{p}^{∞}(∂G ∩ B(a, 2r)) ≥ r^{p}/c_{1}

with c_{1} = c_{1}(c, n, p).

2000 Mathematics Subject Classification: Primary 30C65.

A K -quasiconformal ball satisfies these conditions for 1 ≤ p ≤ n − 2 with c = c(K, n), and (1.1) follows. In the homological version, (1) and (2) are replaced by

(1) H_{n−1}(G) = 0,

(2) the natural homomorphisms H_{k}(G∩B(x, t)) → H_{k}(G∩B(x, ct)) are zero for n − p ≤ k ≤ n − 2.

In a recent paper [BK, 6.5], M. Bonk and P. Koskela proved an inner version of (1.1), where the euclidean ball B(a, 2r) was replaced by a ball in the inner length metric of G (and m_{n−1} by m_{n−1}^{∞}).

The purpose of this paper is to prove inner versions of the results in [Vä]. We find it convenient to replace the Hausdorff content m_{p}^{∞} by the projectional measure μ_{p}, defined in 2.3. Since μ_{p} ≤ m_{p}^{∞} ≤ m_{p}, the new results are somewhat stronger than those in [Vä] also in the euclidean metric.

I thank M. Bonk for calling my attention to this problem.

2. Preliminaries

2.1. Notation. Open balls in R^{n} are written as B(x, r) where x is the center and r is the radius. For a, b ∈ R^{n}, we let [a, b] denote the closed line segment with endpoints a, b, and we set [a, b) = [a, b] \ {b}. For 1 ≤ p ≤ n, the family of all pdimensional linear subspaces of R^{n} is written as G_{p}(R^{n}). For E ∈ G_{p}(R^{n}), we let π_{E}: R^{n} → E denote the orthogonal projection. The distance between nonempty subsets A, B of R^{n} is d(A, B), and the diameter of a set A ⊂ R^{n} is d(A). The volume of the unit ball in R^{n} is α(n).

2.2. Hausdorff measure and content. For A ⊂ R^{n} we let m_{p}(A) denote the Hausdorff (outer) p-measure of A, defined and normalized as in [Fe, 2.10.2], and the Hausdorff p-content m_{p}^{∞}(A) is defined similarly but without any restrictions on the diameters of the covering sets. Then m_{p}^{∞}(A) ≤ m_{p}(A). We consider only the case where p is an integer, 0 ≤ p ≤ n, and the normalization means that m_{p}(A) = m_{p}^{∞}(A) is the Lebesgue (outer) p-measure of A whenever A lies in a p-dimensional affine subspace of R^{n}.

2.3. Projectional measure. Let n ≥ p ≥ 1 be integers. The projectional p-measure of a set A ⊂ R^{n} is the number

μ_{p}(A) = sup{m_{p}(π_{E}A): E ∈ G_{p}(R^{n})}.

Since the diameter of a set is decreased in each π_{E}, we have always

μ_{p}(A) ≤ m_{p}^{∞}(A) ≤ m_{p}(A),

but there are, for example, sets A ⊂ R^{2} with μ_{1}(A) = 0 < m_{1}^{∞}(A); see [Ma, 9.2]. It is easy to see that the set function μ_{p} is monotone and countably subadditive, hence a measure in the sense of [Fe, 2.1.2].

3. Projectional versions of the wall theorems

We show that one can replace the Hausdorff content m_{p}^{∞} by the projectional measure μ_{p} in the wall theorems of [Vä].

3.1. Wall theorem. The homological [Vä, 1.3], homotopical (1.2 and [Vä, 6.2]) and the quasiconformal [Vä, 1.11] wall theorems are true with m_{p}^{∞} replaced by μ_{p}.

Proof. Inspection of the proofs of [Vä] shows that it suffices to prove a projectional version of the Grid lemma [Vä, 2.3]. This is done in 3.2 below. We recall the notation of [Vä, 2.2].

For s > 0 let 𝒦_{s} be the natural decomposition of R^{n} into closed cubes of side s. For 0 ≤ k ≤ n we let K_{s}^{k} denote the union of all k -dimensional faces of the members of 𝒦_{s}.

3.2. Grid lemma. Suppose that 1 ≤ p ≤ n, that s > 0, and that A ⊂ R^{n} is a set with μ_{p}(A) < s^{p}/N, where N is the binomial coefficient \binom{n}{p}. Then there is a ∈ R^{n} such that A ∩ (K_{s}^{n−p} + a) = ∅.

Proof. We may assume that s = 1. Let F be one of the N linear subspaces of R^{n} spanned by p vectors of the standard basis (e_{1}, …, e_{n}). Set J = [0, 1)^{n}, J_{F} = π_{F} J and

D_{F} = ⋃{J_{F} ∩ (π_{F} A − w): w ∈ Z^{n} ∩ F }.

Since the p-cubes J_{F} + w are disjoint and m_{p} measurable, we obtain (see [HS, 10.9])

\begin{aligned}m_{p}(D_{F}) & ≤ ∑_{w} m_{p}(J_{F} ∩ (π_{F} A − w)) = ∑_{w} m_{p}((J_{F} + w) ∩ π_{F} A) \\ & = m_{p}(π_{F} A) ≤ μ_{p}(A) < 1/N. \end{aligned}

Set D_{F}^{∗} = J∩π_{F}^{−1}D_{F} and D = ⋃_{F} D_{F}^{∗} . By Fubini’s theorem we obtain m_{n}(D_{F}^{∗}) < 1/N, and hence m_{n}(D) < 1. Since K_{1}^{n−p} is the union of the sets π_{F}^{−1}[Z^{n} ∩ F], the lemma holds with any a ∈ J \ D.

3.3. Remarks. 1. Lemma 3.2 has three advantages compared with [Vä, 2.3]: (1) Since μ_{p} ≤ m_{p}^{∞}, it is stronger, (2) the constant N is smaller than the corresponding constant β(n, p) of [Vä, 2.3], (3) the proof is simpler.

2. In [Vä, 2.3] there is a misprint: s/β should be s^{p}/β.

4. Wall properties and inner wall properties

4.1. Wall properties. Let c > 0, and let n ≥ 2 and 1 ≤ p ≤ n − 1 be integers. We say that a domain G ⊊ R^{n} has the (c, p)-wall property if

μ_{p}(∂G ∩ B(a, 2r)) ≥ r^{p}/c

whenever a ∈ G and r = d(a, ∂G). Observe that since μ_{p}(B(a, 2r)) = α(p)(2r)^{p}, we always have c ≥ 2^{−p}/α(p).

4.2. Inner wall properties. Let G ⊂ R^{n} be a domain. We let λ = λ_{G} denote the inner length metric of G, defined by

λ(a, b) = inf_{γ} l(γ),

where l(γ) is the length of γ and the infimum is taken over all rectifiable arcs γ joining a and b in G. The distance λ(a, b) is also defined if a ∈ G and b is a boundary point of G, accessible from G by a rectifiable arc. For a ∈ G and r > 0, we let B_{λ}(a, r) denote the set of all x ∈ \overline{G} such that λ(a, x) is defined and λ(a, x) < r. Since λ(a, b) ≤ |a − b|, we always have B_{λ}(a, r) ⊂ B(a, r).

We say that a domain G ⊊ R^{n} has the inner (c, p)-wall property if

μ_{p}(∂G ∩ B_{λ}(a, 2r)) ≥ r^{p}/c

whenever a ∈ G and r = d(a, ∂G). Again c ≥ 2^{−p}/α(p).

Trivially, the (c, p)-wall property implies the inner (c, p)-wall property. The purpose of this section is to prove the converse where, however, c must be replaced by a larger constant c′(c, p). This implies the inner version of the Wall Theorem 3.1, given in 4.9.

4.3. Lemma. Let G ⊊ R^{n} be a domain, let p ∈ [1, n − 1] be an integer and let s ≥ 2. Suppose that

μ_{p}(∂G ∩ B_{λ}(a, sr)) ≥ r^{p}/c

whenever a ∈ G and r = d(a, ∂G). Then G has the inner (c′, p)-wall property with c′ = (s − 1)^{p}c.

Proof. Let a ∈ G and r = d(a, ∂G). Choose a point b ∈ ∂G with |a − b| = r, and set

e = (b − a)/r, t = r/(s − 1), y = b − te.

Then d(y, ∂G) = t. It is easy to see that B_{λ}(y, st) ⊂ B_{λ}(a, 2r). Hence

μ_{p}(∂G ∩ B_{λ}(a, 2r)) ≥ μ_{p}(∂G ∩ B_{λ}(y, st)) ≥ t^{p}/c = r^{p}/c′.

4.4. Theorem. If a domain G ⊊ R^{n} has the (c, p)-wall property, it has the inner (c′, p)-wall property with c′ = c′(c, p).

Proof. Let a ∈ G and r = d(a, ∂G). It suffices to find an estimate

(4.5)  μ_{p}(∂G ∩ B_{λ}(a, sr)) ≥ r^{p}/c_{1},

where c_{1} and s ≥ 2 depend only on c and p, since the theorem then follows from 4.3 with c′ = (s − 1)^{p}c_{1} . We shall prove (4.5) with the universal constant s = \frac{9}{4} and with c_{1} = 4 · 8^{p}c, which yield c′ = 4 · 10^{p}c.

We shall several times make use of the elementary inequality

(4.6)  |π_{E}x| + |x − π_{E}x| ≤ |x|\sqrt{2} ,

valid for all x ∈ R^{n} and for all linear subspaces E ⊂ R^{n}.

We may assume that a = 0. Choose a point b ∈ ∂G with |b| = r, and set

e = b/|b|, t = r/8, y = b − te = 7te, A = ∂G ∩ B(y, 2t).

Since μ_{p}(A) ≥ t^{p}/c, there is a subspace F ∈ G_{p}(R^{n}) such that m_{p}(π_{F} A) ≥ t^{p}/2c. Set π = π_{F} , π′ = π_{F^{⊥}}. By (4.6) we have either |πy| ≤ |y|/\sqrt{2} or |π′y| ≤ |y|/\sqrt{2} .

Case 1: |πy| ≤ |y|/\sqrt{2} . For each x ∈ A we have
|πx| ≤ |πy| + 2t ≤ 7t/\sqrt{2} + 2t < 7t < r,

and hence [0, πx] ⊂ G. Since x ∈ ∂G, there is a point z_{x} ∈ ∂G ∩ [πx, x] with [πx, z_{x}) ⊂ G. By (4.6) we obtain

\begin{aligned}λ(0, z_{x}) & ≤ l([0, πx] ∪ [πx, z_{x}]) ≤ |πx| + |πx − x| ≤ |x|\sqrt{2} \\ & ≤ (|y| + 2t)\sqrt{2} = 9t\sqrt{2} < 2r. \end{aligned}

Setting A_{1} = {z_{x} : x ∈ A} we thus have A_{1} ⊂ B_{λ}(0, 2r) and πA_{1} = πA. Hence

μ_{p}(∂G ∩ B_{λ}(0, 2r)) ≥ m_{p}(πA_{1}) = m_{p}(πA) ≥ t^{p}/2c,

which implies (4.5) with s = 2, c_{1} = 2 · 8^{p}c.

Case 2: |π′y| ≤ |y|/\sqrt{2} . Set q = n − p and
B = B(y, 2t), Z = πB + π′B, D_{x} = πx + π′B

for x ∈ A. Then x ∈ D_{x}, and D_{x} is a q -disk of radius 2t with π′D_{x} = π′B. For z ∈ Z we have

(4.7)  |π′z| ≤ |π′y| + 2t ≤ |y|/\sqrt{2} + 2t = (7/\sqrt{2} + 2)t < 7t < r.

Furthermore, by (4.6) we get

(4.8)  |πz| + |π′z| ≤ |πy| + |π′y| + 4t ≤ |y|\sqrt{2} + 4t < 14t < 2r

for all z ∈ Z.

We consider two subcases, each of which contains two subsubcases.

Subcase 2a: p ≤ q.

Subsubcase 2a1. There is a point x ∈ A such that ∂G meets [w, π′w] for each w ∈ D_{x}. Let w ∈ D_{x}. Then |π′w| < r by (4.7). Now there is a point z_{w} ∈ ∂G ∩ [π′w, w] with [π′w, z_{w}) ⊂ G. By (4.8) we obtain

λ(0, z_{w}) ≤ |π′w| + |π′w − z_{w}| ≤ |π′w| + |π′w − w| < 2r.

Setting D = {z_{w} : w ∈ D_{x}} we have D ⊂ ∂G ∩ B_{λ}(0, 2r), and π′D = π′D_{x} is the q -disk F^{⊥} ∩ B(π′y, 2t).

Since p ≤ q, we can choose a subspace E ∈ G_{p}(R^{n}) with E ⊂ F^{⊥}. Then π_{E}D is a p-disk of radius 2t, and thus

μ_{p}(∂G ∩ B_{λ}(0, 2r)) ≥ m_{p}(π_{E}D) = α(p)2^{p}t^{p}.

Since 2^{p}α(p) ≥ 1/c by 4.1, this gives (4.5) with s = 2, c_{1} = 8^{p}c.

Subsubcase 2a2. For each x ∈ A there is a point w_{x} ∈ D_{x} such that ∂G ∩ [w_{x}, π′w_{x}] = ∅. Then [w_{x}, π′w_{x}] ⊂ G. Since x ∈ ∂G, there is a point z_{x} ∈ ∂G ∩ [x, w_{x}] with [w_{x}, z_{x}) ⊂ G. Since |w_{x} − z_{x}| ≤ |w_{x} − x| ≤ d(D_{x}) = 4t, we get by (4.8)

λ(0, z_{x}) ≤ |π′w_{x}| + |π′w_{x} − w_{x}| + |w_{x} − z_{x}| < 14t + 4t = 18t.

Hence the set A_{1} = {z_{x} : x ∈ A} lies in ∂G ∩ B_{λ}(0, 9r/4). Since πA_{1} = πA, we obtain

μ_{p}(A_{1}) ≥ m_{p}(πA_{1}) = m_{p}(πA) ≥ t^{p}/2c,

and (4.5) follows with s = \frac{9}{4} , c_{1} = 2 · 8^{p}c.

Subcase 2b: p > q. Since

dim(F^{⊥} ∪ {y})^{⊥} = p − 1 ≥ p − q,

we can choose F_{1} ∈ G_{p−q}(R^{n}) with F_{1} ⊂ (F^{⊥} ∪ {y})^{⊥}. Setting E = F^{⊥} + F_{1} we have dim E = p and π_{E}y = π′y.

Recall the notation D_{x} = πx + π′B for x ∈ A. Let C be the union of those D_{x} for which ∂G ∩ [w, π_{E}w] ≠ ∅ for all w ∈ D_{x}. Then C ⊂ Z ⊂ B(y, 2t\sqrt{2} ).

Subsubcase 2b1: m_{p}(π_{E}C) ≥ t^{p}/4c. For each x ∈ C we have
|π_{E}x| ≤ |π_{E}y| + 2t\sqrt{2} ≤ 7t/\sqrt{2} + 2t\sqrt{2} < 8t = r;

hence [0, π_{E}x] ⊂ G. Choose a point z_{x} ∈ ∂G ∩ [π_{E}x, x] such that [π_{E}x, z_{x}) ⊂ G. By (4.6) we obtain

\begin{aligned}λ(0, z_{x}) & ≤ |π_{E}x| + |π_{E}x − z_{x}| ≤ |π_{E}x| + |π_{E}x − x| ≤ |x|\sqrt{2} \\ & ≤ (7t + 2t\sqrt{2} )\sqrt{2} < 14t < 2r. \end{aligned}

Hence the set C_{1} = {z_{x} : x ∈ C} lies in ∂G ∩ B_{λ}(0, 2r). Since

μ_{p}(C_{1}) ≥ m_{p}(π_{E}C_{1}) = m_{p}(π_{E}C) ≥ t^{p}/4c,

we get (4.5) with s = 2, c_{1} = 4 · 8^{p}.

Subsubcase 2b2: m_{p}(π_{E}C) ≤ t^{p}/4c. For v ∈ π_{E}C set Q(v) = π_{E}^{−1}{v} ∩ Z. Since E^{⊥} ⊂ F, we have for each x ∈ Q(v)

|π_{E^{⊥}}x − π_{E^{⊥}}y| ≤ |πx − πy| < 2t.

Thus Q(v) is contained in a q -disk of radius 2t. By Fubini’s theorem we obtain

m_{n}(C) ≤ ∫_{π_{E}C} m_{q}(Q(v)) dm_{p}(v) ≤ α(q)(2t)^{q}m_{p}(π_{E}C) ≤ α(q)(2t)^{q}t^{p}/4c.

On the other hand, Fubini’s theorem also gives

m_{n}(C) = m_{q}(D_{x})m_{p}(πC) = α(q)(2t)^{q}m_{p}(πC),

and hence m_{p}(πC) ≤ t^{p}/4c. Since m_{p}(πA) ≥ t^{p}/2c, we obtain m_{p}(π[A \ C]) ≥ t^{p}/4c.

Let x ∈ A \ C. There is a point w_{x} ∈ D_{x} such that ∂G ∩ [w_{x}, π_{E}w_{x}] = ∅. Since |π_{E}w_{x}| ≤ |π_{E}y| + 3t = |π′y| + 3t < r, we have [w_{x}, π_{E}w_{x}] ⊂ G. Since x ∈ ∂G, there is a point z_{x} ∈ ∂G ∩ [x, w_{x}] with [w_{x}, z_{x}) ⊂ G. Then |w_{x} − z_{x}| ≤ |w_{x} − x| ≤ d(D_{x}) ≤ 4t. By (4.6) we get

\begin{aligned}λ(0, z_{x}) & ≤ |π_{E}w_{x}| + |π_{E}w_{x} − w_{x}| + |w_{x} − z_{x}| \\ & ≤ |w_{x}|\sqrt{2} + 4t ≤ (7 + 2\sqrt{2} )t\sqrt{2} + 4t < 18t. \end{aligned}

Hence the set A_{1} = {z_{x} : x ∈ A \ C} lies in B_{λ}(0, 9r/4). Since

μ_{p}(A_{1}) ≥ m_{p}(π_{F} A_{1}) = m_{p}(π_{F} [A \ C]) ≥ t^{p}/4c,

we get (4.5) with s = 9r/4, c_{1} = 4 · 8^{p}c.

4.9. Inner wall theorem. Suppose that a domain G ⊊ R^{n} satisfies the hypotheses of one of the wall theorems [Vä, 1.3], [Vä, 6.2] (see 1.2) with a constant c > 0 and an integer p ∈ [1, n − 1]. Then G has the inner (c′, p)-wall property with c′ = c′(c, n, p). If G is K -quasiconformally equivalent to a ball, then G has the inner (c′, n − 1)-wall property with c′ = c′(K, n).

Proof. The theorem follows from Theorems 3.1 and 4.4.

444 Jussi Väisälä

References

[BK] Bonk, M., and P. Koskela: Conformal metrics and size of the boundary. - Amer. J. Math. (to appear).

[Fe] Federer, H.: Geometric measure theory. - Springer, 1969.

[He] Heinonen, J.: The boundary absolute continuity of quasiconformal mappings II. - Rev. Mat. Iberoamericana 12, 1996, 697–725.

[HS] Hewitt, E., and K. Stromberg: Real and abstract analysis. - Springer, 1969.

[Ma] Mattila, P.: Geometry of sets and measures in euclidean spaces. - Cambridge University Press, 1995.

[Vä] Väisälä, J.: The wall conjecture on domains in euclidean spaces. - Manuscripta Math. 93, 1997, 515–534.

Received 13 November 2001