Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 19, 1994, 375–392

QUASISYMMETRIC EMBEDDINGS OF PRODUCTS OF CELLS INTO THE EUCLIDEAN SPACE

Pekka Alestalo and Jussi Väisälä

University of Helsinki, Department of Mathematics P.O. Box 4 (Hallituskatu 15), FIN-00014 Helsinki, Finland

Abstract. We shall mainly consider the following situation. Suppose that A is a p-cell, B is a q -cell and there exists a quasisymmetric embedding f: A × B → R^{p+q}. What can we then say about the metric properties of A and B? It turns out that, for example, the cell A locally satisfies a (p − 1)-dimensional bounded turning condition. We also study the packing measures of A and B and show that if B is a quasiconvex arc, this implies certain conditions for the packing measure of A.

Introduction

The motivation for studying quasisymmetric embeddings of products of cells into the Euclidean space of the corresponding dimension derives from the article [Vä_{3} ], where the second author considered the product of two curves and showed that the existence of a QS embedding of the product into the plane implies certain regularity properties of the curves. The purpose of this article is to extend these results to cover some higher dimensional cases.

As regards [Vä_{3} ], it turns out that similar results can be obtained for the cells in our case, but this requires some new concepts that generalize the usual bounded turning and quasiconvexity properties.

The methods used in this article are similar to those in [Vä_{3} ] and [Tu, Lemma 4]. However, some modifications are needed for handling higher dimensional objects; one of these is the use of packing measures. To avoid topological obstructions we restrict our study mainly to products of cells instead of arbitrary manifolds.

1. Metric spaces and manifolds

We start by listing some notation used hereafter.

All spaces considered here will be metric and the distance between points a and b in a space (X, d) is written as |a − b| even if X is not a normed space. If A and B are nonempty subsets of a space X, we let d(A) denote the diameter of A and d(A, B) denote the distance between A and B. Also, we let B(A, r) and \overline{B}(A, r) denote the open and closed neighborhoods of A with radius r.

1991 Mathematics Subject Classification: Primary 30C65.

The product of two metric spaces A and B is the set A × B endowed with the metric

|(a, b) − (a′, b′)| = |a − a′| + |b − b′|,

whenever a, a′ ∈ A and b, b′ ∈ B. If A_{1}, …, A_{n} are spaces, we identify the spaces ∏_{j=1}^{k} A_{j} × ∏_{j=k+1}^{n} A_{j} and ∏_{j=1}^{n} A_{j} for all 1 ≤ k ≤ n − 1.

Let N denote the natural numbers starting from 1, let R denote the set of reals and let I = {x ∈ R | 0 ≤ x ≤ 1}. In the Euclidean n-space R^{n} we use the ordinary metric and let B^{n} = B(0, 1), S^{n−1} = ∂B^{n} and Ω_{n} = m_{n}(B^{n}), where m_{n} is the Lebesgue measure of R^{n}.

A metric space X is said to be boundedly precompact if all bounded subsets of X are precompact. A space X is boundedly compact if all closed and bounded subsets of X are compact.

Using elementary topology, we get the following lemma.

1.1. Lemma. A boundedly compact space is complete and, conversely, a complete and boundedly precompact space is boundedly compact.

A boundedly compact space homeomorphic to R^{p} is called a p-string. It is easy to show that bounded compactness is preserved in products and hence the product of a p-string with a q -string is a (p + q)-string.

A manifold may have a boundary unless this is explicitly denied. A metric space X homeomorphic to the space I^{p} is called a p-cell. The boundary of a p-cell X is the image of ∂I^{p} under a homeomorphism I^{p} → X. A 1-cell is also called an arc. For an arc A, we let s(A, 𝒫) denote the length of the broken line determined by a finite division 𝒫 of A, and let l(A) denote the length of A.

Let X be a p-cell. The inner radius of X is the number

r(X) = max{d(x, ∂X) | x ∈ X}.

Clearly r(X) ≤ d(X) for all cells.

1.2. Lemma. If A is an arc, then r(A) ≥ d(A)/4.

Proof. Let a and b be the end points of A. Then

r(A) = max_{x∈A}{|x − a| ∧ |x − b|},

and the assertion follows from [Vä_{3} , 2.7], which gives a point x ∈ A such that |x − a| ∧ |x − b| ≥ d(A)/4.

Let X be a metric space and let c ≥ 1. The space X is of c-bounded turning, abbreviated c-BT, if, given disjoint points a and b in X, there exists an arc A ⊂ X joining a and b such that d(A) ≤ c|a−b|. The space X is called locally c-BT if every point x ∈ X has a neighborhood U such that all pairs of points in U can be joined by an arc A ⊂ X such that d(A) ≤ c|a − b|. If d(A) is replaced with l(A) in the definition of a BT and locally BT space, we get the properties c-quasiconvex and locally c-quasiconvex. In the literature c-quasiconvex arcs are also said to satisfy the c-chord-arc-condition.

We generalize these concepts as follows.

1.3. Definition. Let c ≥ 1 and let p ≥ 0 be an integer. A metric space A is of (c, p)-bounded turning, abbreviated as (c, p)-BT, if each continuous mapping f: S^{p} → A has a continuous extension g: \overline{B}^{p+1} → A such that d(g\overline{B}^{p+1} ) ≤ cd(fS^{p}). We also say that A is locally (c, p)-BT if every point has a neighborhood U in A such that every continuous mapping f: S^{p} → A with fS^{p} ⊂ U has a continuous extension g: \overline{B}^{p+1} → A satisfying the above condition.

The space A is said to be (c, p)-CBT if, given a (p + 1)-cell β in A, there is a (p + 1)-cell γ ⊂ X such that ∂γ = ∂β and d(γ) ≤ cd(∂γ). The local CBTcondition is defined by requiring every point to have a neighborhood U such that for every (p + 1)-cell whose boundary lies in U there is another (p + 1)-cell in A with the same boundary and satisfying the above turning condition.

Obviously the property (c, 0)-BT is equivalent to the ordinary concept of c-BT defined earlier. Also, a space is (c, 0)-CBT if and only if each of its path components is c-BT.

For example, convex sets in normed spaces are (1, p)-BT for every p ≥ 0.

1.4. Lemma. Let A be a p-cell and let c ≥ 1. If A is (c, p − 1)-BT, it is also (c, p − 1)-CBT.

Proof. Let β = ∂γ for some p-cell γ in A. Then γ is the unique p-cell in A with this property, and we must show that d(γ) ≤ cd(β). Let f: S^{p−1} → A be an embedding such that fS^{p−1} = β. Then it has an extension g: \overline{B}^{p} → A such that d(g\overline{B}^{p} ) ≤ cd(fS^{p−1}) = cd(β), and it suffices to show that γ ⊂ g\overline{B}^{p} . We may assume that A = \overline{B}^{p} . Let h: \overline{B}^{p} → A be an embedding extending f; thus h\overline{B}^{p} = γ. Then h ≃ g rel S^{p−1} so that their degrees agree in int γ. The degree of h is clearly ±1, and therefore g\overline{B}^{p} contains γ.

We do not know whether the converse is true.

The following lemma shows that these properties are preserved in products.

1.5. Lemma. Let A and B be metric spaces, let p ≥ 0 and let c ≥ 1. If A and B are (locally) (c, p)-BT, then A × B is (locally) (2c, p)-BT and, conversely, if A × B is (locally) (c, p)-BT, then both A and B are (locally) (c, p)-BT. For (local) quasiconvexity the same is true even without the factor 2.

Proof. We start with the global case. Let A and B be (c, p)-BT and let f: S^{p} → A × B be continuous. Then pr_{1} ∘ f: S^{p} → A and pr_{2} ∘ f: S^{p} → B are continuous and thus have extensions g_{1}: \overline{B}^{p+1} → A and g_{2}: \overline{B}^{p+1} → B such that d(g_{i}\overline{B}^{p+1} ) ≤ cd(pr_{i}fS^{p}) ≤ cd(fS^{p}), i = 1, 2. Then g = g_{1} × g_{2}: \overline{B}^{p+1} → A × B is an extension of f and

d(g\overline{B}^{p+1} ) ≤ d(g_{1}\overline{B}^{p+1} ) + d(g_{2}\overline{B}^{p+1} ) ≤ 2cd(fS^{p}).

Thus A × B is (2c, p)-BT.

Conversely, suppose that A × B is (c, p)-BT. To show that A is then (c, p)- BT let f: S^{p} → A. Choose an arbitrary point b ∈ B and let f_{1}: S^{p} → A × B be defined by f_{1}(x) = (f(x), b). Then f_{1} has an extension g_{1}: \overline{B}^{p+1} → A × B such that d(g_{1}\overline{B}^{p+1} ) ≤ cd(f_{1}S^{p}) = cd(fS^{p}). Thus g = pr_{1} ∘ g_{1} is the required extension of f. Hence A is c-BT.

Turning to the local case, let A and B be locally (c, p)-BT and let x = (a, b) ∈ A × B. Choose neighborhoods U and V of a and b, respectively, such that the BT-conditions are satisfied. Then U × V is a neighborhood of x, and the same kind of reasoning as above shows it to satisfy the turning condition.

Finally, assume that A×B is locally (c, p)-BT and let a ∈ A. Choose a point b ∈ B and a neighborhood U × V of (a, b), which is contained in a neighborhood satisfying the turning condition. We show that U is the required neighborhood of a. Let f: S^{p} → A be continuous with fS^{p} ⊂ U. Define f_{1}: S^{p} → A × B by setting f_{1}(x) = (f(x), b). Then f_{1}S^{p} ⊂ U × V , and therefore f_{1} has an extension g_{1}: \overline{B}^{p+1} → A × B such that d(g_{1}\overline{B}^{p+1} ) ≤ cd(f_{1}S^{p}). Then g = pr_{1} ∘ g_{1} is an extension of f. As before, it is easily shown that the turning condition is also satisfied for g.

The corresponding proofs for quasiconvexity are similar, if one considers arcs instead of extensions of maps.

We recall the definition of a quasisymmetric mapping from [TV, p. 97]. Let X and Y be metric spaces and let η: [0, ∞[→ [0, ∞[ be a homeomorphism. An embedding f: X → Y is η -quasisymmetric, abbreviated η -QS, if |f(a) − f(x)| ≤ η(t)|f(b) − f(x)| for all points a, b, x ∈ X such that |a − x| ≤ t|b − x|. The inverse of an η -QS embedding f: X → Y is η′ -QS in fX with η′(t) = η^{−1}(t^{−1})^{−1} for t > 0. Hereafter, the expression “f is η -QS” contains the assumption that η is a self-homeomorphism of [0, ∞[. For convenience we shall assume that η(1) ≥ 1.

An arc A is called an η -QS arc if there is an η -QS homeomorphism f: I → A. A metric characterization for quasisymmetric arcs is given in [TV, 4.9].

The following theorem shows that the concepts defined above are QS invariants.

1.6. Theorem. Let X and Y be metric spaces and let f: X → Y be η -QS. If p ≥ 0, c ≥ 1 and the space X is (locally) (c, p)-BT, then fX is (locally) (2η(c), p)-BT. The same is also true for the property CBT.

Proof. Let h: S^{p} → fX be continuous. Then the map f^{−1} ∘ h: S^{p} → X has a continuous extension g′: \overline{B}^{p+1} → X such that d(g′\overline{B}^{p+1} ) ≤ cd(f^{−1}hS^{p}). Define a map g: \overline{B}^{p+1} → fX by setting g = f ∘ g′. Then g is a continuous extension of h and the first inequality of [TV, 2.5] (substituting A ↦ f^{−1}hS^{p}, B ↦ g′\overline{B}^{p+1} and f ↦ f) gives

d(g\overline{B}^{p+1} ) = d(fg′\overline{B}^{p+1} ) ≤ 2η(\frac{d(g′\overline{B}^{p+1} )}{d(f^{−1}hS^{p})})d(ff^{−1}hS^{p}) ≤ 2η(c)d(hS^{p}).

Thus the space fX is (2η(c), p)-BT.

The proof of the local version is similar, and so is the invariance of the CBTproperty.

Here are some applications of this result.

1.7. Example. If A is a 2-manifold with a spire (see [GV, 10.2]), there exists no QS embedding of A into R^{2} .

In fact, a neighborhood of the spire cannot satisfy the local (c, 1)-BT-condition for any c ≥ 1, and the claim follows from the theorem.

Let K ≥ 1 and n ≥ 2. An (n − 1)-manifold M ⊂ R^{n} is K -quasicon-formally locally flat if for every x ∈ M there is a neighborhood V of x and a K -quasiconformal map f: (V, V ∩ M) → (B^{n}, B^{n−1} × {0}).

A domain D ⊂ Ṙ^{n} is called a K -quasiball if there is a K -quasiconformal map f: (Ṙ^{n}, B^{n}) → (Ṙ^{n}, D).

1.8. Theorem. Let M be an (n − 1)-manifold in R^{n} with n ≥ 2. If K ≥ 1 and if M is K -quasiconformally locally flat, then M is locally (c(K), p)-BT for every p ≥ 0.

Proof. Let x ∈ M and let U be a neighborhood of x in R^{n} such that (U, U ∩ M) is equivalent to (B^{n}, B^{n−1} × {0}) via a K -QC map f: U → B^{n}. There is a neighborhood V ⊂ U containing x such that fV = B(0, r) for some r > 0 and f | V is η -QS with η depending only on K. Then f^{−1} | B(0, r) is η′ - QS, and B^{n−1}(0, r)×{0} is (1, p)-BT for all p ≥ 0. Thus V ∩M is a neighborhood of x satisfying for all p the local (c, p)-BT-condition with c = c(K).

1.9. Corollary. If G ⊂ R^{n} is a K -quasiball, then at every finite point ∂G is locally (c(K), p)-BT for every p ≥ 0.

2. Packing measures

In this section we define the packing measure in a metric space. It was introduced by C. Tricot in [Tr] for Euclidean spaces and further studied in [TT] and [SRT]. In a metric space it was defined by D. Sullivan in [Su, Section 8] and also by H. Haase in [Ha]. We give a description of the definition in [TT] applied to a metric space, which is a special case of Haase’s more general version, and prove some of its properties.

Let X be a metric space. Furthermore, let E be a nonempty subset of X, let J ⊂ N and let δ > 0. Suppose that for every index j ∈ J a point x_{j} ∈ E and a number r_{j} are given such that 0 < r_{j} ≤ δ and that |x_{i} − x_{j}| > r_{i} + r_{j} for all i ≠ j. Then we say that the collection 𝒫 = {B(x_{j}, r_{j})}_{j∈J} is a δ -packing of E, and if p > 0, we set s_{p}(𝒫) = ∑_{j∈J}(2r_{j})^{p}, τ_{δ}^{p}(∅) = 0 and

τ_{δ}^{p}(E) = sup {s_{p}(𝒫) | 𝒫 is a δ-packing of E} .

The p-dimensional packing premeasure of E is the number

τ^{p}(E) = inf_{δ>0} τ_{δ}^{p}(E) = lim_{δ→0+} τ_{δ}^{p}(E).

With this definition τ^{p} is the same as φ − P of [TT] and P^{φ} of [SRT] if X = R^{n} and φ(t) = t^{p}.

The set function τ^{p} is not an outer measure, since it fails to be countably subadditive. For example, every unbounded set has an infinite p-dimensional packing premeasure for every p > 0. Using Munroe’s Method I the p-dimensional packing measure ν^{p}(E) of E is defined by

ν^{p}(E) = inf{∑_{k∈N} τ^{p}(E_{k}) | F ⊂ ⋃_{k∈N}E_{k}}.

Clearly ν^{p} ≤ τ^{p}, and ν^{p} is the same as φ − p of [TT] and p^{φ} of [SRT], φ as above.

A δ -covering of a set E ⊂ X is a countable collection {B_{i}}_{i∈I} of subsets of X such that E ⊂ ∪_{i∈I}B_{i} and d(B_{i}) ≤ δ for all i ∈ I. The p-dimensional Hausdorff measure ℋ ^{p}(E) of E is defined by

ℋ ^{p}(E) = lim_{δ→0+} inf{∑_{i∈N} d(B_{i})^{p} | {B_{i}}_{i∈N} is a δ-covering of E}.

For our purposes the following theorem gives the most important properties of the packing measure ν^{p}.

2.1. Theorem. The set function ν^{p} is a regular metric outer measure and corresponds to a countably additive Borel measure. It has the following additional properties.

1. Always ν^{p} ≥ ℋ ^{p}.

2. In R^{p} we have Ω_{p}ν^{p} = 2^{p}m_{p}.

3. If p > 0, M ≥ 0, f: X → Y is M -Lipschitz and A ⊂ X, then τ^{p}(fA) ≤ M^{p}τ^{p}(A) and ν^{p}(fA) ≤ M^{p}ν^{p}(A).

4. If A is an arc, then ν^{1}(A) = τ^{1}(A) = l(A).

5. If A_{1}, …, A_{p} are arcs, then

\frac{2^{p}}{Ω_{p}} ∏_{i=1}^{p} l(A_{i}) ≤ ν^{p}( ∏_{i=1}^{p} A_{i}) ≤ \frac{2^{p}p^{p/2}}{Ω_{p}} ∏_{i=1}^{p} l(A_{i}).

Proof. By [Mu, 11.3] Method I gives an outer measure. That ν^{p} is metric is proved as in [TT, 5.1].

Since ν^{p} is a metric outer measure, all Borel sets are measurable and [Mu, 12.3.1] implies that ν^{p} is regular, since in the definition of ν^{p} we may restrict our attention to closed covers by virtue of the equation τ^{p}( \overline{E} ) = τ^{p}(E). So the five properties remain to be proved.

1. The proof of this is similar to the corresponding one in R^{n} as presented in [SRT, 3.3], once we have a counterpart for their Lemma 2.1 in metric spaces. This is given in Lemma 2.2 below.

Let E′ ⊂ X. It suffices to show that ℋ ^{p}(E) ≤ τ^{p}(E) for all E ⊂ X, since given this, ℋ ^{p}(E′) ≤ ∑_{I∈N} ℋ ^{p}(E_{i}) ≤ ∑_{i∈N} τ^{p}(E_{i}) for every covering {E_{i}}_{i∈N} of E′, and this implies that ℋ ^{p}(E′) ≤ ν^{p}(E′).

We may assume that ℋ ^{p}(E) > 0 and τ^{p}(E) < ∞. Let H < ℋ ^{p}(E), let ε > 0 and choose δ_{1} > 0 such that H ≤ ∑_{i∈N} d(B_{i})^{p} + ε for every δ_{1} -covering {B_{i}}_{i∈N} of E. Then choose δ_{2} > 0 such that s_{p}(𝒫) ≤ τ^{p}(E) + ε for all δ_{2} - packings 𝒫 of E. Let δ = (δ_{1}/6) ∧ δ_{2} , and choose, by Lemma 2.2 below, a δ -packing 𝒫 = {B(x_{i}, r_{i})}_{i∈I} of E satisfying the condition in the lemma. Then ∑_{i∈I}(2r_{i})^{p} ≤ τ^{p}(E) + ε. For each j the collection

ℬ_{j} = \{\overline{B}(x_{i}, r_{i}) | 1 ≤ i ≤ j\} ∪ \{\overline{B}(x_{i}, 3r_{i}) | i > j\}

is a δ_{1} -covering of E and therefore

H ≤ ∑_{i=1}^{j}(2r_{i})^{p} + ∑_{i>j}(6r_{i})^{p} + ε ≤ τ^{p}(E) + 2ε + 3^{p} ∑_{i>j}(2r_{i})^{p}.

The last series converges, it is at most 3^{p}s_{p}(𝒫), and approaches zero as j tends to infinity. This implies that H ≤ τ^{p}(E) + 2ε. Since ε > 0 and H < ℋ ^{p}(E) were arbitrary, this gives ℋ ^{p}(E) ≤ τ^{p}(E), proving the claim.

2. If E ⊂ R^{p}, the inequality 2^{p}m_{p}(E) ≤ Ω_{p}ν^{p}(E) follows from the first property. To prove the converse for τ^{p}, let A ⊂ R^{p} be a cube, perhaps containing some of its boundary. Let δ > 0 and let 𝒫 = {B(x_{j}, r_{j})}_{j∈J} be a δ -packing of A. Then ⋃_{j∈J} B(x_{j}, r_{j}) ⊂ B(A, δ). It follows that

Ω_{p}s_{p}(𝒫) = 2^{p} ∑_{j∈J} m_{p}(B(x_{j}, r_{j})) = 2^{p}m_{p}(∪_{j∈J}B(x_{j}, r_{j})) ≤ 2^{p}m_{p}(B(A, δ)),

and, further, that Ω_{p}ν^{p}(A) ≤ Ω_{p}τ^{p}(A) ≤ Ω_{p}τ_{δ}^{p}(A) ≤ 2^{p}m_{p}(B(A, δ)) for all δ. This implies that Ω_{p}ν^{p}(A) ≤ 2^{p}m_{p}(A). An arbitrary open set U ⊂ R^{p} can be expressed as a disjoint union of such cubes, and therefore Ω_{p}ν^{p}(U) ≤ 2^{p}m_{p}(U). If E ⊂ R^{p} is arbitrary, then Ω_{p}ν^{p}(E) ≤ Ω_{p}ν^{p}(U) ≤ 2^{p}m_{p}(U) for all open sets U containing E, and thus Ω_{p}ν^{p}(E) ≤ 2^{p}m_{p}(E).

3. For the first inequality it suffices to show that τ_{δ}^{p}(fA) ≤ M^{p}τ_{δ/M}^{p}(A) for all δ > 0. So let δ > 0 and let 𝒫 = {B(x_{j}, r_{j})}_{j∈J} be a δ -packing of fA. For each j ∈ J we choose a point y_{j} ∈ f^{−1}(x_{j}). Then 𝒫_{1} = {B(y_{j}, r_{j}/M)}_{j∈J} is a δ/M -packing of E and thus s_{p}(𝒫) = M^{p}s_{p}(𝒫_{1}) ≤ M^{p}τ_{δ/M}^{p}(A). Since 𝒫 was arbitrary, the inequality follows.

For the second claim we may assume that ν^{p}(A) < ∞. Let {E_{k}}_{k∈N} cover A such that ∑_{k} τ^{p}(E_{k}) < ∞. Setting F_{k} = fE_{k} for each k we get a covering {F_{k}}_{k∈N} of fA with ν^{p}(fA) ≤ ∑_{k} τ^{p}(F_{k}) ≤ M^{p} ∑_{k} τ^{p}(E_{k}), where the first part was used. This implies the second inequality.

4. The inequality l(A) ≤ ν^{1}(A) follows from the first property, since ℋ ^{1} equals the length for arcs. Because of the inequality ν^{1} ≤ τ^{1} , it suffices to show that τ^{1}(A) ≤ l(A). We may assume that l(A) < ∞. Let λ = l(A) and let f: [0, λ] → A be the parametrization of A with respect to arc length; then f is 1-Lipschitz. By the third property it follows that

τ^{1}(A) = τ^{1}(f[0, λ]) ≤ τ^{1}([0, λ]) = λ,

where the last equality follows as in the proof the the second property.

5. To simplify notation we only prove the claim for two arcs, and consider first the latter inequality. We may assume that λ_{i} = l(A_{i}) < ∞ for i = 1, 2. For both i let f_{i}: [0, λ_{i}] → A_{i} be the parametrization with respect to the arc length; thus the maps f_{i} are 1-Lipschitz. Define a map h: [0, λ_{1}] × [0, λ_{2}] → A_{1} × A_{2} by setting h(x, y) = (f_{1}(x), f_{2}(y)). Then we have

\begin{aligned}|h(x, y) − h(x′, y′)| & = |f_{1}(x) − f_{1}(x′)| + |f_{2}(y) − f_{2}(y′)| ≤ |x − x′| + |y − y′| \\ & ≤ \sqrt{2}\sqrt{|x − x′|^{2} + |y − y′|^{2}} \end{aligned}

so that h is \sqrt{2}-Lipschitz. From this it follows by the second and third properties that

\begin{aligned}ν^{2}(A_{1} × A_{2}) & = ν^{2}(h([0, λ_{1}] × [0, λ_{2}])) ≤ 2ν^{2}([0, λ_{1}] × [0, λ_{2}]) \\ & = \frac{2 · 2^{2}}{Ω_{2}} m_{2}([0, λ_{1}] × [0, λ_{2}]) = \frac{8}{π} λ_{1}λ_{2}. \end{aligned}

For the converse, let K_{i} < λ_{i} and choose divisions 𝒫_{1} of A_{1} with points x_{0}, …, x_{n} and 𝒫_{2} of A_{2} with points y_{0}, …, y_{n} such that s(A_{i}, 𝒫_{i}) ≥ K_{i}, i = 1, 2. Let α_{i} = A_{1}[x_{i−1}, x_{i}[ and β_{i} = A_{2}[y_{i−1}, y_{i}[, 1 ≤ i ≤ n. Define maps g_{ij}: α_{i} × β_{j} → R^{2} by setting g_{ij}(x, y) = (|x − x_{i−1}|, |y − y_{j−1}|). Then we have

\begin{aligned}|g_{ij}(x, y) − g_{ij}(x′, y′)| & = \sqrt{(|x − x_{i−1}| − |x′ − x_{i−1}|)^{2} + (|y − y_{j−1}| − |y′ − y_{j−1}|)^{2}} \\ & ≤ \sqrt{|x − x′|^{2} + |y − y|^{2}} ≤ |x − x′| + |y − y′|, \end{aligned}

so that each g_{ij} is 1-Lipschitz. Therefore

K_{1}K_{2} ≤ s(A_{1}, 𝒫_{2})s(A_{2}, 𝒫_{2}) = ∑_{i,j=1}^{n} |x_{i} − x_{i−1}||y_{j} − y_{j−1}|

≤ ∑_{i,j=1}^{n} m_{2}(g_{ij}(α_{i} × β_{j})) = \frac{π}{2^{2}} ∑_{i,j=1}^{n} ν^{2}(g_{ij}(α_{i} × β_{j})) ≤ \frac{π}{4} ∑_{i,j=1}^{n} ν^{2}(α_{i} × β_{j})

= \frac{π}{4} ν^{2}(⋃_{i,j=1}^{n}α_{i} × β_{j}) ≤ \frac{π}{4} ν^{2}(A_{1} × A_{2}).

Since this holds for all K_{i} < l(A_{i}), i = 1, 2, we get the first inequality. The proof is now complete. Had we used the Euclidean metric in the product of the arcs, the first inequal-

ity of the fifth property would hold as an equality, since all the maps g_{ij} and also h would be 1-Lipschitz.

In the proof of the first property we used the following lemma, which replaces [SRT, 2.1].

2.2. Lemma. Let ∅ ≠ E ⊂ X and let δ > 0 be such that τ_{δ}^{p}(E) < ∞. Then there is a δ -packing 𝒫 = {B(x_{i}, r_{i})}_{i∈I} of E, where either I = {1, …, n} or I = N, satisfying for each j ∈ I the condition

E \ ⋃_{i=1}^{j}\overline{B}(x_{i}, r_{i}) ⊂ ⋃_{i>j}\overline{B}(x_{i}, 3r_{i}).

Proof. Let x_{1} ∈ E be arbitrary and set r_{1} = δ. Inductively, if B(x_{j}, r_{j}) is already chosen and E_{j} = E \ ⋃_{i=1}^{j} \overline{B}(x_{i}, r_{i}) ≠ ∅, we choose a point x_{j+1} ∈ E_{j} with

|x_{j+1} − x_{i}| − r_{i} > \frac{3}{4} sup_{y∈E} min\{|y − x_{l}| − r_{l} | 1 ≤ l ≤ j\} = a_{j+1}

for every 1 ≤ i ≤ j, and let r_{j+1} = δ∧a_{j+1} ; the condition τ_{δ}^{p}(E) < ∞ implies E to be bounded, and therefore a_{j+1} is finite. Then the collection 𝒫 = {B(x_{i}, r_{i})}_{i∈I} is a δ -packing of E. Since τ_{δ}^{p}(E) < ∞, the series s_{p}(𝒫) converges and thus either I = {1, …, n} for some n ≥ 1 or else I = N and lim_{i→∞} r_{i} = 0. In the former case we have E ⊂ ∪_{i=1}^{n}\overline{B}(x_{i}, r_{i}), and the claimed property is obvious.

Suppose then that the latter case occurs. Let j ∈ N and let x ∈ E_{j}. Then ε = d(x, ∪_{i=1}^{j}\overline{B}(x_{i}, r_{i})) > 0, and by the construction we can choose the smallest index k > j with |x − x_{k}| ≤ r_{k} + ε ∧ δ. We claim that x ∈ \overline{B}(x_{k}, 3r_{k}). This is clear if r_{k} = δ, and we may assume that r_{k} < δ, i.e. r_{k} = a_{k}. If x ∉ \overline{B}(x_{k}, 3r_{k}), then 3r_{k} < |x − x_{k}| ≤ r_{k} + ε ∧ δ, and hence ε ∧ δ > 2r_{k}. Therefore, for some 1 ≤ s ≤ k − 1

r_{k} ≥ \frac{3}{4}(|x − x_{s}| − r_{s}) > \frac{3}{4}(ε ∧ δ + r_{s} − r_{s}) > \frac{3}{2}r_{k},

a contradiction. Thus the claimed property is proved.

The following regularity property is connected with packing measures of balls in a metric space.

2.3. Definition. Let p > 0 and let A be a metric space. We say that A is c-bounded in p-measure if for all x ∈ A and r > 0 the inequality

ν^{p}(B(x, r)) ≤ c (2r)^{p}

is valid.

For example, an open set in R^{p} is 1-bounded in p-measure, and it is easy to construct examples of manifolds without this property. A c-quasiconvex arc A is c-bounded in 1-measure, but not conversely. In fact, given x ∈ A and r > 0, let y and z be the first and last points of A in \overline{B}(x, r) in some orientation. Then by Theorem 2.1 we have

ν^{1}(B(x, r)) ≤ ν^{1}(A[y, z]) = l(A[y, z]) ≤ c|y − z| ≤ 2cr.

3. Turning conditions

In this section we study the BT-properties of the cells. The main result is Theorem 3.4.

3.1. Theorem. Let A be an m-manifold and B an n-manifold, both without boundary. If f: A × B → R^{m+n} is η -QS, then A and B are locally (c, p)-BT for all p ≥ 0 with the constant c depending only on η.

Proof. Using the invariance of domain for manifolds we see that the set f(A × B) is open and hence locally (1, p)-BT. By Lemma 1.6 the manifold A × B is locally (2η′(1), p)-BT. Now Lemma 1.5 implies the result.

The following theorem gives more information in a special case, although the conditions are quite restrictive.

3.2. Theorem. Let A, B and Y be metric spaces and let f: A × B → Y be η -QS. If c ≥ 1 and f(A × B) is (c, p)-BT for some p ≥ 0, then both A and B are (c_{1}, p)-BT with c_{1} = c_{1}(c, η).

Proof. By Lemma 1.5 it is enough to show that A × B is (c_{1}, p)-BT, which follows from Theorem 1.6.

3.3. Theorem. Let A be an m-string, let B be an n-string and let f: A × B → R^{m+n} be η -QS. Then f is surjective, and for all p ≥ 0, A and B are (c, p)-BT with c = c_{η}.

Proof. By Theorem 3.2 it suffices to show that f is surjective. This is the case, since by invariance of domain f(A×B) is open, and by Lemma 1.1 and [TV, 2.24] it is complete and hence also closed.

The following theorem is a generalization of [Vä_{3} , 3.2].

3.4. Theorem. Let A be a p-cell, let B be a q -cell and let f: A×B → R^{p+q} be η -QS. Let H′ = η^{−1}(1)^{−1} and assume that d(A) < r(B)/H′. Then A is (c, p − 1)-CBT with c = c(η). If p = 1, we can choose c = 2H′.

Proof. Given a p-cell A′ in A, we must show that d(A′) ≤ cd(∂A′). We may assume that A′ = A, since the assumptions hold for A′ as well.

Suppose first that the (p + q)-cell f(A × B) is locally flat in R^{p+q}. Choose points a_{0}, a_{1} ∈ ∂A with |a_{1} − a_{0}| = d(∂A), let b_{0} ∈ B be such that B(b_{0}, r(B)) ∩ ∂B = ∅, and let y ∈ ∂B. For each b ∈ B set A(b) = A × {b} and let z_{i} = (a_{i}, b_{0}), i = 0, 1. If z ∈ A(b_{0}) and z′ ∈ A(y), then

|z − z_{0}| ≤ d(A) < \frac{r(B)}{H′} ≤ \frac{1}{H′} |y − b_{0}| ≤ \frac{1}{H′} |z′ − z_{0}|.

By the quasisymmetry of f we conclude that |f(z) − f(z_{0})| ≤ |f(z′) − f(z_{0})|. Let

r = sup{|f(z) − f(z_{0})| | z ∈ A(b_{0})}

and G = B(f(z_{0}), r) ⊂ R^{p+q}. Then fA(b_{0}) ⊂ \overline{G}. We claim that the intersection G ∩ f(A × ∂B) is empty. In fact, if (a, b) ∈ A × ∂B, then the choice y = b and z′ = (a, b) above leads to the inequality |f(a, b) − f(z_{0})| ≥ |f(z) − f(z_{0})|, valid for all z ∈ A(b_{0}). Hence |f(a, b) − f(z_{0})| ≥ r and the claim follows. Let C = A(b_{0}), ∂C = ∂A × {b_{0}}, and let g: C → R^{p+q} be an extension of f | ∂C defined by g = k ∘ h, where h: C → \overline{B}^{p} is a homeomorphism such that h(z_{0}) = e_{1} , and k: \overline{B}^{p} → R^{p+q} is the conical extension of f ∘ h^{−1} | S^{p−1} with vertices e_{1} and f(z_{0}). Then g is continuous and gC ⊂ \overline{G}, and therefore g ≃ f | C rel ∂C in R^{p+q} \ f(A × ∂B). By Lemma 3.6 below we have pr_{1}f^{−1}gC = A.

If u ∈ f^{−1}gC, then

(1)  |f(u) − f(z_{0})| ≤ d(gC) ≤ d(fC) ≤ 2η(1)|f(z_{1}) − f(z_{0})|,

where the last inequality follows from [TV, 2.5], since d(C) = |z_{1} − z_{0}|. This implies that |u − z_{0}| ≤ η′(2η(1))|z_{1} − z_{0}| = η′(2η(1))|a_{1} − a_{0}|, and therefore d(A) ≤ d(f^{−1}gC) ≤ 2η′(2η(1))|a_{1} − a_{0}|. If p = 1, then gC is a line segment and obviously (1) is valid without the factor 2η(1). This proves the case where f(A × B) is locally flat in R^{p+q}.

If the flatness assumption is not valid, we proceed by approximating the cell A × B from within and using the previous result as follows. Let φ: I^{p} ≈ A, let ψ: I^{q} ≈ B and let ε > 0. Choose a number s ∈]0, 1/2[ such that if A_{1} = ϕ[s, 1 − s]^{p}, then d(A) ≤ d(A_{1}) + ε and d(∂A_{1}) ≤ d(∂A) + ε. Then choose t ∈]0, 1/2[ such that r(ψ([t, 1 − t]^{q})) > H′d(A_{1}) and let B_{1} = ψ([t, 1 − t]^{q}). The cell f(A_{1} × B_{1}) is then locally flat in R^{p+q}, and it follows from the first part of the proof that d(A_{1}) ≤ cd(∂A_{1}), and further that

d(A) ≤ d(A_{1}) + ε ≤ cd(∂A_{1}) + ε ≤ cd(∂A) + (c + 1)ε.

Since ε > 0 was arbitrary, the theorem follows from this.

3.5. Corollary. Let A_{1}, …, A_{p} be arcs and let f: ∏_{i=1}^{p} A_{i} → R^{p} be η -QS. If H′ = η^{−1}(1)^{−1} and

d(A_{1}) < \frac{1}{4H′} min_{2≤i≤p} d(A_{i}),

then A_{1} is 2H′ -BT.

Proof. By Lemma 1.2 we have r(∏_{k>1} A_{k}) = min{r(A_{k}) | k > 1} ≥ min{d(A_{k})/4 | k > 1} so that d(A_{1}) < r(∏_{k>1} A_{k})/H′, and the assertion follows from the theorem.

The following topological result was used in the proof of Theorem 3.4.

3.6. Lemma. Let A and B be as in Theorem 3.4 and let f: A × B → R^{n} be an embedding, n = p + q. Let b_{0} ∈ int B, let C = A × {b_{0}} and let g: C → R^{n} be a continuous map such that g ≃ f | C rel ∂C in R^{n} \ f(A × ∂B), where ∂C = ∂A × {b_{0}}. If the n-cell Q = f(A × B) is locally flat in R^{n}, then pr_{1}f^{−1}gC = A.

Proof. We may assume that A = \overline{B}^{p} , B = \overline{B}^{q} and b_{0} = 0. Using the local flatness of Q and the Shoenflies theorem we can extend f to R^{n} and further reduce the situation to the case where f = id_{A×B}. Then Q = A × B and we must show that P = pr_{1}(gC ∩ A × (B \ ∂B)) = A. To get a contradiction, assume that a_{0} ∈ A \ P. Since ∂C is fixed by g, we have ∂A ⊂ P, and may thus assume that a_{0} = 0. Let V = Ṙ^{n} \ (A × ∂B) and choose a retraction r: V → Ṙ^{p} × {0} such that r^{−1}(0, 0) = {0} × int B; in fact, the map r can be defined by setting r(∞) = ∞, and for (x, y) ∈ V \ {∞},

r(x, y) = \{ \begin{matrix}∞ & if |y| ≥ 1, \\ (x, 0) & if |y| < 1 and |x| ≤ 1, \\ \frac{|x| − |y|}{|x|(1 − |y|)}(x, 0) & if |y| < 1 and |x| > 1. \end{matrix}

Define a map g_{1}: A → Ṙ^{p} by the compositions

A⟶^{i}A × {0}⟶^{g}V ⟶^{r}Ṙ^{p} × {0}⟶^{pr_{1}}Ṙ^{p},

where i is the embedding x ↦ (x, 0). Since g ≃ f | C rel ∂C, we have g_{1} ≃ i_{A} rel ∂A, where i_{A}: A ↪ Ṙ^{p}. Therefore the degree of g_{1} in int A is equal to the degree of i_{A}, which implies that g_{1}A ⊃ A. This is impossible, because 0 ∈ A \ P and r^{−1}(0, 0) = {0} × int B imply that 0 ∉ g_{1}A.

Therefore P = A and the lemma is proved.

The following example, which is a three-dimensional version of [Vä_{3} , 3.4], shows that the condition involving the inner radius of B cannot be omitted in the previous theorem.

3.7. Example. Let A = S^{1} , let B = I^{2} and let f: A × B → R^{3} be defined by f(e, (s, t)) = (1 + t)e − se_{3} , where e_{3} = (0, 0, 1). Elementary calculations show that f is 2\sqrt{2}-bilipschitz and hence η -QS with η(t) = 8t. However, there is no upper bound for the BT- and quasiconvexity constants of subarcs of A; the condition d(A) < r(B)/H′ is not satisfied, since d(A) = 2, r(B) = 1/2 and H′ = 8.

4. Rectifiability conditions

In this section we study the rectifiability of the embedded manifolds dealing with quasiconvexity and packing measures. We start with the following generalization of [Vä_{3} , 4.2].

4.1. Theorem. Let A be an arc, let B be a p-manifold without boundary and let f: A × B → R^{p+1} be η -QS. Let a_{0} and a_{1} be the end points of A, let B_{0} ⊂ B be open and assume that

β = inf{|f(a_{0}, b) − f(a_{1}, b)| | b ∈ B_{0}} > 0.

Then we have

ℋ ^{p}(B_{0}) ≤ ν^{p}(B_{0}) ≤ c(η, p)\frac{m_{p+1}(fQ)}{β^{p+1}}l(A)^{p},

where Q = int(A × B_{0}).

Proof. The first inequality follows from Theorem 2.1. To prove the second one, we may assume that A is rectifiable and that m(fQ) < ∞. Suppose first that \overline{B}_{0} is compact in B. Let 0 < δ < d(A) be such that \overline{B}(\overline{B}_{0}, δ) is a compact subset of B, and let 𝒫 = {B(x_{j}, r_{j})}_{j∈J} be a δ -packing of B_{0} . Fix j ∈ J and choose a division P = {y_{0}, y_{1}, …, y_{m(j)}} of A such that μ_{i} = |y_{i} − y_{i−1}| ∈ [r_{j}/2, r_{j}] for all 1 ≤ i ≤ m(j). Set A_{i} = A[y_{i−1}, y_{i}] for 1 ≤ i ≤ m(j). By Theorem 3.4 we may assume δ to be small enough so that the arcs A_{i} are c_{1} -BT with c_{1} = c_{1}(η).

Let 1 ≤ i ≤ m(j). Choose a point y ∈ A_{i} such that |y −y_{i}| = |y −y_{i−1}| ≡ μ′. Then we have μ′ ∈ [r_{j}/4, c_{1}r_{j}]. We set z = (y, x_{j}), z′ = (y_{i−1}, x_{j}), z′′ = (y_{i}, x_{j}) and β_{ij} = |f(z′′) − f(z′)|, and let Q_{ij} = int A_{i} × B(x_{j}, r_{j}). Choose a point u ∈ ∂Q_{ij} such that |f(u) − f(z)| = d(f(z), ∂(fQ_{ij})) ≡ r. If μ′ ≤ r_{j}, then μ′ = μ′ ∧ r_{j} ≤ |u − z|; otherwise μ′ ≤ c_{1}r_{j} = c_{1}(r_{j} ∧ μ′) ≤ c_{1}|u − z|. This latter inequality is therefore valid in both cases and, consequently, |z′ − z| = |y − y_{i−1}| = μ′ ≤ c_{1}|u − z|. Since f is η -QS, it follows that |f(z′) − f(z)| ≤ η(c_{1})r. Similarly it can be shown that |f(z′′) − f(z)| ≤ η(c_{1})r. These inequalities imply that β_{ij} ≤ 2η(c_{1})r. As an open set fQ_{ij} is measurable, and we have m_{p+1}(fQ_{ij}) ≥ m_{p+1}(B(f(z), r)) = Ω_{p+1}r^{p+1} , and hence β_{ij}^{p+1} ≤ c_{2}m_{p+1}(fQ_{ij}),

where c_{2} = (2η(c_{1}))^{p+1}/Ω_{p+1} . Using the Hölder inequality we get

\begin{aligned}β^{p+1} & ≤ ( ∑_{i=1}^{m(j)} β_{ij})^{p+1} ≤ ( ∑_{i=1}^{m(j)} β_{ij}^{p+1})( ∑_{i=1}^{m(j)} 1)^{p} = m(j)^{p} ∑_{i=1}^{m(j)} β_{ij}^{p+1} \\ & ≤ c_{2}m(j)^{p}m_{p+1}(fQ_{j}), \end{aligned}

where Q_{j} = int A × B(x_{j}, r_{j}).

Now, r_{j} ≤ 2μ_{i} for all indices 1 ≤ i ≤ m(j), which implies that

m(j)r_{j} ≤ 2 ∑_{i=1}^{m(j)} μ_{i} ≤ 2l(A).

Therefore,

s_{p}(𝒫) = 2^{p} ∑_{j∈J} r_{j}^{p} ≤ \frac{2^{2p}c_{2}}{β^{p+1}} l(A)^{p} ∑_{j∈J} m_{p+1}(fQ_{j}) ≤ c(η, p)\frac{m_{p+1}(fQ)}{β^{p+1}}l(A)^{p},

where c(η, p) = 4^{p}c_{2} .

Since the collection 𝒫 is an arbitrary δ -packing of B_{0} , we have ν^{p}(B_{0}) ≤ τ^{p}(B_{0}) ≤ τ_{δ}^{p}(B_{0}) ≤ c(η, p)m_{p+1}(fQ)l(A)^{p}/β^{p+1} .

Now let B_{0} be an arbitrary open set in B. By the previous considerations we have ν^{p}(B_{1}) ≤ c(η, p)m_{p+1}(fQ)l(A)^{p}/β^{p+1} for all open sets B_{1} such that \overline{B}_{1} is a compact subset of B_{0} , and this implies that the above inequality also holds for ν^{p}(B_{0}).

Theorem 4.1 can be regarded as a generalisation of the following special case. Let A = [a, b] ⊂ R be an interval, let B ⊂ R^{p} be a bounded domain, let f: A × B → R^{p+1} be η -QS in the Euclidean metric of A × B and let β be as in Theorem 4.1. Let Γ be the path family of horizontal line segments connecting the sets {a} × B and {b} × B in Q = A × B ⊂ R^{p+1} . By [Vä_{1} , 7.3] the modulus of Γ is m_{p}(B)/l(A)^{p} and by [Vä_{1} , 7.1] the modulus of the image-family satisfies M(fΓ) ≤ m_{p+1}(fQ)/β^{p+1} . Since f is η(1)^{p} -quasiconformal in int Q, the inequality M(Γ) ≤ η(1)^{p}M(fΓ) implies the conclusion of 4.1 with c = 2^{p}η(1)^{p}/Ω_{p}.

4.2. Theorem. Let A, B and f be as in Theorem 4.1 and suppose that l(A) ≥ d(B). If c ≥ 1 and A is c-quasiconvex, then B is c_{1} -bounded in pmeasure with c_{1} = c_{1}(c, η, p).

Proof. Let x ∈ B and let r > 0. We may assume that r ≤ d(B). Choose points a_{0} and a_{1} in A such that l(A[a_{0}, a_{1}]) = r. Then r ≤ c|a_{1} − a_{0}|. Setting A_{0} = A[a_{0}, a_{1}] and B_{0} = B(x, r) we have by Theorem 4.1

ν^{p}(B_{0}) ≤ c_{2} \frac{m_{p+1}(fQ)}{β^{p+1}}l(A_{0})^{p} = c_{2}r^{p} \frac{m_{p+1}(fQ)}{β^{p+1}},

where Q = A_{0} × B_{0} , β = inf{|f(a_{0}, b′) − f(a_{1}, b′)| | b′ ∈ B_{0}} and c_{2} = c_{2}(η, p). To prove the theorem it thus suffices to find an upper bound for m_{p+1}(fQ)/β^{p+1} .

Let β_{1} > β and let b_{0} ∈ B_{0} be such that |f(a_{0}, b_{0}) − f(a_{1}, b_{0})| ≤ β_{1} . Set z_{i} = (a_{i}, b_{0}), i = 0, 1. If z = (a, b) ∈ A_{0} × B_{0} , then

|z − z_{0}| = |a − a_{0}| + |b − b_{0}| ≤ l(A_{0}) + 2r = 3r ≤ 3c|a_{1} − a_{0}| = 3c|z_{1} − z_{0}|,

and hence |f(z) − f(z_{0})| ≤ η(3c)|f(z_{1}) − f(z_{0})| ≤ η(3c)β_{1} . From this it follows that m_{p+1}(fQ) ≤ Ω_{p+1}η(3c)^{p+1}β_{1}^{p+1} , and letting β_{1} → β we get the theorem with c_{1} = c_{2}2^{−p}Ω_{p+1}η(3c)^{p+1} .

4.3. Remarks. 1. The assumption of local quasiconvexity of A in the above theorem will be quite natural after Corollary 4.7 below.

2. Suppose that in Theorem 4.2 the manifold B is the interior of a p-cell B_{1} such that d(B_{1}) < d(A)/4H′, where H′ = η^{−1}(1)^{−1} . Then combining Lemma 1.2 and Theorems 3.4 and 4.2 we may conclude that

ν^{p}(B′) ≤ c_{2}d(∂B′)^{p}

for all p-cells B′ ⊂ B and for some c_{2} = c_{2}(c, η, p).

Next we turn our attention to the product of more than two arcs. The case of two arcs is contained in Theorem 4.1 by Theorem 2.1.

4.4. Theorem. Let A_{1}, …, A_{p} be arcs, let f: ∏_{k=1}^{p} A_{k} → R^{p} be η -QS, let a_{1} and b_{1} be the end points of A_{1} and let

β_{1} = min{|f(a_{1}, y) − f(b_{1}, y)| | y ∈ ∏_{k=2}^{p} A_{k}}.

Then

∏_{k=2}^{p} l(A_{k}) ≤ c(p, η)\frac{m_{p}(fQ)}{β_{1}^{p}}l(A_{1})^{p−1},

where Q = ∏_{k=1}^{p} A_{k}.

Proof. This immediately follows from Theorem 4.1 and the fifth property in 2.1.

This makes it possible to prove the local quasiconvexity of all the arcs provided that one of them is quasiconvex.

4.5. Theorem. Let A_{1}, …, A_{p} be arcs, let f: ∏_{k=1}^{p} A_{k} → R^{p} be η -QS and let H′ = η^{−1}(1)^{−1} . Suppose that c ≥ 1, that A_{1} is c-quasiconvex and

d(A_{2}) < min_{k≠2} d(A_{k})/4H′.

Then A_{2} is c_{1} -quasiconvex with c_{1} = c_{1}(c, η, p).

Proof. It follows from Theorem 4.4 that all the arcs are rectifiable. Let a_{2} and b_{2} be distinct points in A_{2} . We must show that l(A_{2}[a_{2}, b_{2}]) ≤ c_{1}|a_{2} − b_{2}| with a suitable constant c_{1} = c_{1}(c, η, p). Since d(A_{k}) > d(A_{2}) for all k ≠ 2, we can choose points a_{1} and b_{1} in A_{1} such that |a_{1} − b_{1}| = |a_{2} − b_{2}|, and for all 3 ≤ k ≤ p points a_{k} and b_{k} in A_{k} such that d(A_{k}[a_{k}, b_{k}]) = |a_{2} − b_{2}|. Set A′_{k} = A_{k}[a_{k}, b_{k}] and A′ = ∏_{k=1}^{p} A′_{k}. By Theorem 4.4 we have

∏_{k=2}^{p} l(A′_{k}) ≤ c_{2} \frac{m_{p}(fA′)}{β_{1}^{p}}l(A′_{1})^{p−1},

where c_{2} = c_{2}(η, p) and β_{1} is as in Theorem 4.4.

Choose a point x_{1} ∈ ∏_{k=2}^{p} A′_{k} such that β_{1} = |f(a_{1}, x_{1}) − f(b_{1}, x_{1})| and set z_{0} = (a_{1}, x_{1}) and z_{1} = (b_{1}, x_{1}). From Corollary 3.5 it follows that the arc A_{2} is 2H′ -BT. If z ∈ A′, then

|z − z_{0}| ≤ d(A′_{1}) + d(A′_{2}) + ⋯ + d(A′_{p})
≤ l(A′_{1}) + 2H′|a_{2} − b_{2}| + |a_{2} − b_{2}| + ⋯ + |a_{2} − b_{2}|
≤ c|a_{1} − b_{1}| + 2H′|a_{2} − b_{2}| + (p − 2)|a_{2} − b_{2}|
= c_{3}|a_{1} − b_{1}| = c_{3}|z_{0} − z_{1}|,

where c_{3} = 2H′ +c+p−2. Hence |f(z)−f(z_{0})| ≤ η(c_{3})|f(z_{0})−f(z_{1})| = η(c_{3})β_{1} , and so m_{p}(fA′) ≤ Ω_{p}η(c_{3})^{p}β_{1}^{p}. Setting c_{4} = c_{2}Ω_{p}η(c_{3})^{p} we get the inequality

∏_{k=2}^{p} l(A′_{k}) ≤ c_{4}l(A′_{1})^{p−1},

so that

\begin{aligned}l(A′_{2}) & ≤ c_{4} \frac{l(A′_{1})^{p−1}}{∏_{k=3}^{p} l(A′_{k})} ≤ c_{4}c^{p−1} \frac{|a_{1} − b_{1}|^{p−1}}{∏_{k=3}^{p} d(A′_{k})} = c_{4}c^{p−1} \frac{|a_{2} − b_{2}|^{p−1}}{|a_{2} − b_{2}|^{p−2}} \\ & = c_{4}c^{p−1}|a_{2} − b_{2}|, \end{aligned}

from which the theorem follows with c_{1} = c_{4}c^{p−1} .

It is necessary to set an upper bound for the diameter of A_{2} in the previous theorem; otherwise we cannot get even a global BT-condition, as is shown in Example 3.7 above.

The following example shows that it is indeed possible to embed a product of locally non-rectifiable arcs quasisymmetrically into the Euclidean space.

4.6. Example. In [Tu, p. 151–152] P. Tukia gives an example of a homeomorphism f: R → J, where J is a BT but locally non-rectifiable 1-string of the snow-flake type. He also shows that for all x, y ∈ R the double inequality

\frac{1}{M} |x − y|^{α} ≤ |f(x) − f(y)| ≤ M|x − y|^{α}

is true for some constant M ≥ 1 and α = ln 3/ ln 4. From this it follows that f is η -QS with η(t) = M^{2}t^{α}. Generalizing this, it is easy to show that the mapping F = f × ⋯ × f: R^{n} → J^{n} is θ-QS with θ(t) = Mnt^{α}, for example. Hence F^{−1}: J^{n} → R^{n} is θ′ -QS with θ′(t) = θ^{−1}(t^{−1})^{−1} for t > 0.

In [Tu, Lemma 4] Tukia proved that no QS embedding of A × I^{p} into R^{p+1} exists if A is a non-rectifiable arc. The following corollary is a generalisation of this and also summarizes the case of several arcs considered above.

4.7. Corollary. Let A_{1}, …, A_{p} be arcs and let f: ∏_{k=1}^{p} A_{k} → R^{p} be η -QS. Then the following assertions are true.

1. All the arcs are locally c-BT with c = c(η).

2. If some arc has a rectifiable subarc, all the arcs are rectifiable.

3. If some arc has a c-quasiconvex subarc, all the arcs are locally c_{1} -quasiconvex with c_{1} = c_{1}(c, η, p).

We close this article by raising the following questions.

4.8. Open problems. 1. Let A be a p-cell and suppose that it is (c, p−1)- CBT for some c ≥ 1. Is it then also (c_{1}, p − 1)-BT for some c_{1} = c_{1}(c), or even for c_{1} = c?

2. In Theorems 4.1 and 4.2 one of the factors is an arc. Is it also possible to obtain similar results for the product of manifolds of arbitrary dimensions?

3. In this last section, the dimension of the measure is always an integer. Can one also prove similar results for other measures?

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392 Pekka Alestalo and Jussi Väisälä

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Received 22 March 1993