LIPSCHITZ CELL DECOMPOSITION

IN O-MINIMAL STRUCTURES

Wiesław Pawłucki

Uniwersytet Jagielloński, Instytut Matematyki ul. Reymonta 4, 30-059 Kraków, Poland

December 1, 2006

Abstract. A main tool in studying topological properties of sets definable in ominimal structures is the Cell Decomposition Theorem. This paper proposes its metric counterpart.

1.Introduction.

Fix any o-minimal structure on a real closed field R (for the definition and fundamental properties of o-minimal structures the reader is referred to [vdD]). Let n be a positive integer.

A subset S of R^{n} will be called an (open) cell in R^{n} iff

(1.1)  S = {(x′, x_{n}) ∈ R^{n} : x′ ∈ Δ, φ_{1}(x′) < x_{n} < φ_{2}(x′)},

where x′ = (x_{1}, …, x_{n−1}), Δ is an open definable subset of R^{n−1}, every φ_{i} (i ∈ {1, 2}) is either a definable continuous function φ_{i} : Δ ⟶ R or φ_{i} ≡ −∞ or φ_{i} ≡ +∞ and, for each x′ ∈ Δ, φ_{1}(x′) < φ_{2}(x′).

For any positive M ∈ R, a definable continuous function φ : Δ ⟶ R defined on an open subset Δ of R^{n−1} will be called an M-function iff

(1.2)  |\frac{∂φ}{∂x_{j}} (a)| ≤ M  (j ∈ \{1, …, n − 1\}),

at each point a ∈ Δ in a neighborhood of which φ is of class C^{1}.

An cell S in R^{n} will be called an M-cell (a semi-M-cell) iff, for each i ∈ {1, 2} (for at least one i ∈ {1, 2}), if φ_{i} is finite, it is an M-function. A cell S in R^{n} will be called a regular M-cell iff it is any open interval in the case n = 1 and, in the case n > 1, for each i ∈ {1, 2}, if φ_{i} is finite it is an M-function of class C^{1} on Δ and the projection Δ of S into R^{n−1} is a regular M-cell in R^{n−1}.

2000 Mathematics Subject Classification: Primary 32B20, 14P10. Secondary 32S60, 51N20, 51F99.

Key words and phrases: M-cell, semi-M-cell, M-disc, Lipschitz mapping, cell decomposition.

Research partially supported by the grant of the Polish Ministry of Research and Education 0325/PO3/2004/27 and the European Community IHP-Network RAAG (HPNR-CT-2001-00271)

An M-cell will be called an M-disc iff it is any open interval in the case n = 1 and, in the case n > 1, the both φ_{i} (i ∈ {1, 2}) are finite and admit continuous extensions

(1.3)  φ_{i} : \overline{Δ} ⟶ R

onto the closure of Δ in R^{n−1}, and

(1.4)  φ_{1} = φ_{2} on ∂Δ.

Proposition 1. Let S be a regular M-cell in R^{n} and let φ : S ⟶ R be an Lfunction ( L > 0 ) of class C^{1}.

Then

(1) for any two different points a, b ∈ S, there is a definable continuous mapping

λ = (λ_{1}, …, λ_{n}) : [0 , |a − b|] ⟶ S

such that λ(0) = a, λ(|a − b|) = b and |λ′_{j}(t)| ≤ (j − 1)!M^{j−1}, for any j ∈ {1, …, n} and any t such that λ′_{j}(t) exists^{1};

(2) for any two points a, b ∈ S,

|φ(a) − φ(b)| ≤ n!M^{n−1}L|a − b|.

Proof. (1) Let S be as in (1.1). Arguing by induction and assuming that a′ ≠ b′, one can find a mapping

ω = (ω_{1}, …, ω_{n−1}) : [0 , |a′ − b′|] ⟶ Δ

such that ω(0) = a′, ω(|a′ − b′|) = b′ and |ω′_{j}(τ)| ≤ (j − 1)!M^{j−1}, for any j ∈ {1, …, n − 1} and any τ such that ω′_{j}(τ) exists. Let ε > 0 be such that

φ_{1}(ω(τ)) + ε < φ_{2}(ω(τ)) − ε, for any τ ∈ [0 , |a′ − b′|],

and

φ_{1}(a′) + ε < a_{n} < φ_{2}(a′) − ε and φ_{1}(b′) + ε < b_{n} < φ_{2}(b′) − ε.

Now, it suffices to put

λ_{j}(t) = ω_{j}(t\frac{|a′ − b′|}{|a − b|}), for j ∈ \{1, …, n − 1\},

and

λ_{n}(t) = max\{φ_{1}(ω(t\frac{|a′ − b′|}{|a − b|})) + ε, min\{φ_{2}(ω(t\frac{|a′ − b′|}{|a − b|})) − ε, a_{n} + t\frac{b_{n} − a_{n}}{|a − b|} \}\}.

(2) follows from (1), by the Mean Value Theorem (see [vdD, Chapter 7, (2.3)]).

1|a − b| = \sqrt{∑_{j=1}^{n}(a_{j} − b_{j})^{2}}

Kurdyka-Parusiński Theorem ([K, P]). Any open definable subset G of R^{n} has a finite decomposition

G = S_{1} ∪ ⋯ ∪ S_{k} ∪ Σ,

where every S_{ν} is a regular M_{n}-cell in some linear coordinate system in R^{n} and Σ is nowhere dense, M_{n} being a constant depending only on n.

The aim of the present article is to show that in fact permutations of coordinates are sufficient in the above theorem. We will prove simultaneously by induction on n the following three theorems.

Theorem 1_{n} (2_{n}, 3_{n}). Any open definable subset G of R^{n} has a finite decomposition

(1.5)  G = S_{1} ∪ ⋯ ∪ S_{k} ∪ Σ,

where every S_{ν} is an M_{1n}-cell (M_{2n}-disc, a regular M_{3n}-cell) in R^{n} after a permutation of coordinates and Σ is nowhere dense, M_{1n} (M_{2n}, M_{3n}) being a constant ≥ 1 depending only on n.

For simplicity we will often skip the adjective definable, when considering subsets of spaces R^{n} and mappings between such subsets. Also, we adopt the following conventions. A local property (w) of a mapping f : A ⟶ R^{m}, where A ⊂ R^{n}, is said to be satisfied almost everywhere iff there is a closed subset E of A such that dim E < dim A and (w) is satisfied at each point of A \ E. A finite sequence B_{1}, …, B_{k} of subsets of a set A ⊂ R^{n} is said to be an almost decomposition of A iff B_{ν} (ν = 1, …, k) are pairwise disjoint and dim (A \ (B_{1} ∪ ⋯ ∪ B_{k})) < dim A. This will be denoted by writing

G ≃ B_{1} ∪ ⋯ ∪ B_{k}.

Since Theorem 2_{n} easily implies both Theorems 1_{n} and 3_{n}, it suffices to derive first Theorem 1_{n} from Theorem 2_{n−1} and then Theorem 2_{n} from Theorems 1_{n}, 2_{n−1} and 3_{n−1}. From now on, we will assume that n ≥ 2 is fixed.

2. A preparation.

Lemma 1. If G ⊂ R^{n−1} is open and E ⊂ ∂G is closed of dimension < n − 2 and Theorem 2_{n−1} is true, then G has an almost decomposition

G ≃ Δ_{1} ∪ ⋯ ∪ Δ_{p},

where every Δ_{ν}, after a permutation of coordinates in R^{n−1}, is an M_{2n−1}-disc:

Δ = {(x′′, x_{n−1}) : x′′ ∈ Ω, σ_{ν}(x′′) < x_{n−1} < ρ_{ν}(x′′)}^{2},

such that the both (graphs of )^{3} σ_{ν} and ρ_{ν} are disjoint with E.

2x′′ = (x_{1}, …, x_{n−2})

3We will identify functions with their graphs.

4 WIESŁAW PAWŁUCKI

Proof. Take the projections

π_{j} : R^{n−1} ∋ (x_{1}, …, x_{n−1}) ↦ (x_{1}, …, x_{j−1}, x_{j+1}, …, x_{n−1}) ∈ R^{n−2}, for j ∈ {1, …, n − 1}, and set

Z = the closure of ⋃_{j} π_{j}^{−1}(π_{j}(E)).

Then dim Z ≤ n − 2 and it suffices to use Theorem 2_{n−1} to G \ Z.

As a corollary one easily gets (see [vdD]) the following Lemma 2. If G ⊂ R^{n−1} is open and φ : G ⟶ R is continuous, then G has an almost decomposition

G ≃ Δ_{1} ∪ ⋯ ∪ Δ_{p} ,

where every Δ_{ν}, after a permutation of coordinates in R^{n−1}, is an M_{2n−1}-disc

Δ_{ν} = {(x′′, x_{n−1}) : x′′ ∈ Ω, σ_{ν}(x′′) < x_{n−1} < ρ_{ν}(x′′)}

such that φ|Δ_{ν} has a continuous extension

φ_{ν} : Δ_{ν} ∪ σ_{ν} ∪ ρ_{ν} ⟶ \overline{R} = R ∪ \{−∞, +∞\}

such that φ_{ν}(σ_{ν}) ⊂ R or φ_{ν}(σ_{ν}) = {−∞}, or φ_{ν}(σ_{ν}) = {+∞} and the same for ρ_{ν}.

Proposition 2. Let f : S ⟶ R be a definable C^{1}-function defined on a cell

S = {(x′, x_{n}) ∈ R^{n} : x′ ∈ Δ, φ(x′) < x_{n} < ψ(x′)}

in R^{n} such that φ : Δ ⟶ R is of class C^{1}.

Assume that \frac{∂f}{∂x_{n}} has a finite limit value^{4} at (almost) each point of φ ( for example, when \frac{∂f}{∂x_{n}} is bounded ).

Then there is a closed nowhere dense subset Z of φ such that f extends to a C^{1}-function

f : S ∪ (φ \ Z) ⟶ R

to S ∪ (φ \ Z) as a C^{1}-submanifold with boundary. Proof. It is left to the reader as an exercise (cf [vdD]).

4An element α ∈ \overline{R} is a limit value of a function g : S ⟶ R at a ∈ \overline{S} iff there is an arc γ : (0, 1) ⟶ S such that lim_{t→0} γ(t) = a and lim_{t→0} g(γ(t)) = α.

Lemma 3. Let L, M, N, P ∈ R be positive and let

G = {(x′, x_{n}) : x′ ∈ Δ, φ_{1}(x′) < x_{n} < φ_{2}(x′)}

be a semi-M-cell in R^{n} such that Δ is an N-cell in R^{n−1}, φ_{i} : Δ ⟶ R, for each i ∈ {1, 2},and the following conditions are satisfied almost everywhere in Δ:

(2.1)  |\frac{∂φ_{1}}{∂x_{j}} | ≤ M, for each j ∈ \{1, …, n − 1\};

(2.2)  |\frac{∂φ_{1}}{∂x_{n−1}} | < L < |\frac{∂φ_{2}}{∂x_{n−1}} |;

(2.3)  \begin{matrix}| & \frac{∂φ_{2}}{∂x_{j}} & | \\ | & \frac{∂φ_{2}}{∂x_{n−1}} & | \end{matrix} ≤ P, for each j ∈ \{1, …, n − 1\};

(2.4)  sgn \frac{∂φ_{2}}{∂x_{n−1}} = const.

Then G admits an almost decomposition

G ≃ S_{1} ∪ ⋯ ∪ S_{k},

where every S_{ν} is an M̃-cell, possibly after transposition (x_{n−1}, x_{n}), where M̃ is a positive constant depending only on L, M, N and P.

Proof. Put

Δ = {(x′′, x_{n−1}) : x′′ ∈ Ω, σ(x′′) < x_{n−1} < ρ(x′′)}.

One can assume that

(2.5)  \frac{∂φ_{2}}{∂x_{n−1}} > 0 ;

the other case will follow by a modification. Because of (2.2) and (2.5), it is clear that σ : Δ ⟶ R. By a subdivision of Ω one can assume that σ is of class C^{1} and that (2.2) is satisfied almost everywhere on every segment {(x′′, x_{n−1}) : σ(x′′) < x_{n−1} < ρ(x′′)}, where x′′ ∈ Ω and that φ_{i} admit continuous extensions

φ_{i} : Δ ∪ σ ⟶ R (i = 1, 2)

and

φ_{2} : Δ ∪ ρ ⟶ R ∪ {+∞}

such that φ_{2}(ρ) ⊂ R or φ_{2}(ρ) = {+∞}.

6 WIESŁAW PAWŁUCKI

By Proposition 2, φ_{1} is of class C^{1} almost everywhere on σ. Put

ψ(x′′, x_{n−1}) = φ_{1}(x′′, σ(x′′)) + L(x_{n−1} − σ(x′′)), for (x′′, x_{n−1}) ∈ Δ. Then ψ is an max(M + MN + LN, L)-function and φ_{1} < ψ < φ_{2}.

Now G ≃ S_{1} ∪ S_{2}, where S_{1} = {(x′, x_{n}) : φ_{1}(x′) < x_{n} < ψ(x′)} and S_{2} = {(x′′, x_{n−1}, x_{n}) : x′′ ∈ Ω, Φ_{1}(x′′, x_{n}) < x_{n−1} < Φ_{2}(x′′, x_{n})}, where

Φ_{2}(x′′, x_{n}) = \{ \begin{matrix}ψ^{−1}(x′′, x_{n}) = L^{−1}(x_{n} − φ_{1}(x′′, σ(x′′))) + σ(x′′), \\ if φ_{1}(x′′, σ(x′′)) < x_{n} < ψ(x′′, ρ(x′′)) \\ ρ(x′′), if ψ(x′′, ρ(x′′)) ≤ x_{n} < φ_{2}(x′′, ρ(x′′)) \end{matrix}

and

Φ_{1}(x′′, x_{n}) = \{ \begin{matrix}σ(x′′), & if φ_{1}(x′′, σ(x′′)) < x_{n} ≤ φ_{2}(x′′, σ(x′′)) \\ φ_{2}^{−1}(x′′, x_{n}), & if φ_{2}(x′′, σ(x′′)) < x_{n} < φ_{2}(x′′, ρ(x′′)), \end{matrix}

where ψ^{−1} and φ_{2}^{−1} stand for inversions with respect to x_{n−1}. Lemma 4. Let A ⊂ R^{n−1} be open and let M ∈ R, M > 0. Let f_{α} : A ⟶ R (α ∈ {1, …, k + l}) be M-functions on A each of which has a continuous extension to \overline{A}:

f_{α} : \overline{A} ⟶ R.

Assume that for each a ∈ ∂A there are α ≤ k and β > k such that f_{β}(a) ≤ f_{α}(a).

Then the set

S = {(x′, x_{n}) ∈ A × R : max_{1≤α≤k} f_{α}(x′) < x_{n} < min_{k<β≤k+l} f_{β}(x′)}

is an M-disc in R^{n}. Proof. Indeed,

S = {(x′, x_{n}) ∈ B × R : max_{1≤α≤k} f_{α}(x′) < x_{n} < min_{k<β≤k+l} f_{β}(x′)},

where B is the natural projection of S to A. It is clear that max_{1≤α≤k} f_{α} = min_{k<β≤k+l} f_{β} on ∂B and the lemma follows. Lemma 5. Let α_{1}, α_{2} ∈ \overline{R}, α_{1} < α_{2} and let f, g, h : (α_{1}, α_{2}) ⟶ R be three continuous definable functions such that

(2.6)  g ≤ f on (α_{1}, α_{2});
(2.7)  for each i ∈ {1, 2}, if α_{i} ∈ R, then lim_{t→α_{i}} g(t) = lim_{t→α_{i}} h(t) ∈ R;

(2.8)  sgnf′(t) = const almost everywhere in (α_{1}, α_{2}),

and there is ϵ > 0 such that

(2.9) |f′(t)| ≥ |g′(t)| + ϵ and |f′(t)| > |h′(t)| almost everywhere in (α_{1}, α_{2}).

Then h < f on (α_{1}, α_{2}).

Proof. One can assume that f′(t) > 0. Then α_{1} ∈ R, since otherwise by (2.9), lim_{t→−∞}(f(t) − g(t)) = −∞, a contradiction with (2.6). By (2.9), f − h is strictly

increasing and, by (2.6) and (2.7),

lim_{t→α_{1}}(f(t) − h(t)) ≥ lim_{t→α_{1}}(g(t) − h(t)) = 0.

Hence, f − h > 0 on (α_{1}, α_{2}).

3. Reduction of Theorem 1_{n} to a special case of semi-M-cells.

By the standard cell decomposition theorem (see [vdD]) and since

R^{n} = ⋃_{j=1}^{n}{(x_{1}, …, x_{n}) ∈ R^{n} : |x_{k}| ≤ |x_{j}|, for any k ≠ j},

it suffices to derive Theorem 1_{n} for any cell G in R^{n} such that

(3.1)  G = {(x′, x_{n}) : x′ ∈ Δ, φ_{1}(x′) < x_{n} < φ_{2}(x′)},

where φ_{i} : Δ ⟶ R (i = 1, 2) are continuous.

For given positive L, P ∈ R such a cell G will be called an (L, P)-cell (with respect to the variable x_{r}), where r ∈ {1, …, n − 1}, iff

(3.2)  |\frac{∂φ_{i}}{∂x_{r}} | ≥ L and \begin{matrix}| & \frac{∂φ_{i}}{∂x_{j}} & | \\ | & \frac{∂φ_{i}}{∂x_{r}} & | \end{matrix} ≤ P ,

almost everywhere on Δ, for i ∈ {1, 2}, j ∈ {1, …, n − 1}.

Proposition 3.

(1) Any open cell G ⊂ R^{n} has an almost decomposition

(3.3)  G ≃ S_{1} ∪ ⋯ ∪ S_{k},

where every S_{ν} is either a semi-M_{n}-cell or an (L_{n}, P_{n})-cell after a permutation of coordinates, where positive constants M_{n}, L_{n} and P_{n} depend only on n.

(2) If a cell G is an (L, P)-cell, then G has an almost decomposition (3.3) with only semi-M-cells, where a constant M depends only on n, L and P.

To prove Proposition 3 we first have the following Lemma 6. Let H be an open subset of R^{n} and let E be a closed subset of ∂H such that dim E < n − 1. Let r_{i} ∈ {1, …, n − 1} (i ∈ {1, 2}). Assume that L, P ∈ R are positive and such that, for each a ∈ ∂H \ E:

(3.4 − i) there exists a neighborhood U of a in R^{n} such that ∂H ∩ U is (the graph of) a C^{1}-function ψ : V ⟶ R defined on an open V ⊂ R^{n−1} and such that

|\frac{∂ψ}{∂x_{r_{i}}} | ≥ L and \begin{matrix}| & \frac{∂ψ}{∂x_{j}} & | \\ | & \frac{∂ψ}{∂x_{r_{i}}} & | \end{matrix} ≤ P on V for j ∈ \{1, …, n − 1\},

for i = 1 or i = 2.

Then:

(1) H admits an almost decomposition

(3.5)  H ≃ S_{1} ∪ ⋯ ∪ S_{k},

where every S_{nu} is either a semi-max(L^{−1}, P)-cell or a (P^{−1}, max(L^{−1}, P))- cell in R^{n} after transposition (x_{r_{i}}, x_{n}).

(2) If r_{1} = r_{2} = r, H has such an almost decomposition (3.5) that every S_{ν} is a max(L^{−1}, P)-cell after transposition (x_{r}, x_{n}).

Proof of Lemma 6. After transposition (x_{r_{1}}, x_{n}) take a C^{1}-cell decomposition compatible with each of the sets

Λ_{i} = {a ∈ ∂H \ E : a satisfies (3.4 − i)}

(i = 1, 2) and with E. This gives an almost decomposition

H ≃ S_{1} ∪ ⋯ ∪ S_{k},

where every cell S_{ν} is of the form

S_{ν} = {(x′, x_{n}) : x′ ∈ Δ_{ν}, φ_{1ν}(x′) < x_{n} < φ_{2ν}(x′)},

such that, for i ∈ {1, 2}, either φ_{iν} ⊂ Λ_{1} or φ_{iν} ⊂ Λ_{2}, or φ_{iν} ≡ −∞, or φ_{iν} ≡ +∞.

One can assume that for each i either φ_{iν} ⊂ Λ_{1} or φ_{iν} ⊂ Λ_{2}, since otherwise S_{ν} is trivially a semi-max(L^{−1}, P)-cell.

If φ_{iν} ⊂ Λ_{1}, for at least one i, then S_{ν} is a semi-max(L^{−1}, P)-cell.

If φ_{iν} ⊂ Λ_{2}, for each i ∈ {1, 2}, and r_{1} ≠ r_{2}, then it is easy to check that S_{ν} is an (L, max(L^{−1}, P))-cell with respect to x_{r_{2}}.

Proof of Proposition 3. One can assume that G is as in (3.1). The proof will be by descending induction on the number

⟨G⟩ = ∑_{i=1}^{2} ♯\{j : |\frac{∂φ_{i}}{∂x_{j}} | < 1 + 2M_{2n−1} almost everywhere on Δ\}.

If ⟨G⟩ = 2(n − 1), G is a (1 + 2M_{2n−1})-cell, so assume that ⟨G⟩ < 2(n − 1). Observe that if Δ̃ ⊂ Δ is open, then for G̃ = G ∩ (Δ̃ × R), ⟨G̃⟩ ≥ ⟨G⟩. Hence, one can assume that every φ_{i} is C^{1} and

(3.6)  for each j ∈ \{1, …, n − 1\}, sgn\frac{∂φ_{i}}{∂x_{j}} = const on Δ;

(3.7)  for each j ∈ \{1, …, n − 1\}, either |\frac{∂φ_{i}}{∂x_{j}} | < 1 + 2M_{2n−1}

or |\frac{∂φ_{i}}{∂x_{j}} | > 1 + 2M_{2n−1}, or |\frac{∂φ_{i}}{∂x_{j}} | = 1 + 2M_{2n−1} on Δ

and there is r_{i} ∈ {1, …, n − 1} such that

(3.8)  for each j ∈ \{1, …, n − 1\}, |\frac{∂φ_{i}}{∂x_{j}} | ≤ |\frac{∂φ_{i}}{∂x_{r_{i}}} | on Δ.

Moreover, one can assume that

(3.9)  |\frac{∂φ_{i}}{∂x_{r_{i}}} | ≥ 4M_{2n−1}(1 + 2M_{2n−1}),  for i ∈ \{1, 2\},

since otherwise G is a semi-4M_{2n−1}(1 + 2M_{2n−1})-cell. Besides, by Lemma 2, one can assume that

Δ = {(x′′, x_{n−1}) : x′′ ∈ Ω, σ(x′′) < x_{n−1} < ρ(x′′)}

is an M_{2n−1}-disc and every φ_{i} has a continuous extension

φ_{i} : Δ ∪ σ ∪ ρ ⟶ \overline{R}

such that φ_{i}(σ) ⊂ R or φ_{i}(σ) = {−∞} or φ_{i}(σ) = {+∞}, and the same for ρ.

Observe that if

\frac{∂φ_{1}}{∂x_{n−1}} ▪ \frac{∂φ_{2}}{∂x_{n−1}} ≤ 0,

then clearly G is a semi-M_{2n−1}-cell after transposition (x_{n−1}, x_{n}), so without any loss of generality one can assume that

\frac{∂φ_{i}}{∂x_{n−1}} > 0 on Δ, for i ∈ \{1, 2\}.

By (3.7), one can distinguish the following three cases:

(3.10)  |\frac{∂φ_{i}}{∂x_{n−1}} | ≤ 1 + 2M_{2n−1}, for i ∈ \{1, 2\};

(3.11)  |\frac{∂φ_{i}}{∂x_{n−1}} | ≥ 1 + 2M_{2n−1}, for i ∈ \{1, 2\};

(3.12) |\frac{∂φ_{1}}{∂x_{n−1}} | < 1 + 2M_{2n−1} and |\frac{∂φ_{2}}{∂x_{n−1}} | > 1 + 2M_{2n−1} (or vice-versa).

Case (3.10 ) Here we will be using only that every φ_{i} : Δ ∪ σ ∪ ρ ⟶ R is continuous and there is a closed nowhere dense Z ⊂ Δ such that φ_{i} is C^{1} on Δ \ Z and

(3.13)  |\frac{∂φ_{i}}{∂x_{n−1}} | ≤ 1 + 2M_{2n−1}, on Δ \ Z;

(3.14)  |\frac{∂φ_{i}}{∂x_{j}} | ≤ 3|\frac{∂φ_{i}}{∂x_{r_{i}}} | on Δ \ Z  (j = 1, …, n − 1)

and

(3.15)  |\frac{∂φ_{i}}{∂x_{r_{i}}} | ≥ 2M_{2n−1}(1 + 2M_{2n−1}) on Δ \ Z.

Put

H = {(x′′, x_{n−1}, x_{n}) ∈ G : φ_{2}(x′′, σ(x′′)) < x_{n} < φ_{1}(x′′, ρ(x′′))} =
{(x′, x_{n}) ∈ R^{n} : x′ ∈ D, Φ_{1}(x′) < x_{n} < Φ_{2}(x′)},

where

D = {(x′′, x_{n−1}) ∈ Δ : φ_{2}(x′′, σ(x′′)) < φ_{1}(x′′, ρ(x′′))},
Φ_{1}(x′′, x_{n−1}) = max(φ_{2}(x′′, σ(x′′)), φ_{1}(x′′, x_{n−1}))

and

Φ_{2}(x′′, x_{n−1}) = min(φ_{2}(x′′, x_{n−1}), φ_{1}(x′′, ρ(x′′))).

Observe that Φ_{1} = Φ_{2} on (∂D) ∩ (Δ ∪ σ ∪ ρ), so almost everywhere on ∂D. Besides, if φ_{2}(x′′, σ(x′′)) ≢ −∞, we have by Proposition 2

\frac{∂}{∂x_{j}} φ_{2}(x′′, σ(x′′)) = \frac{∂φ_{2}}{∂x_{j}} (x′′, σ(x′′)) + \frac{∂φ_{2}}{∂x_{n−1}} (x′′, σ(x′′))\frac{∂σ}{∂x_{j}} (x′′),

almost everywhere on Ω, for j ∈ {1, …, n − 2}. Hence, by (3.13) and (3.15)

|\frac{∂}{∂x_{j}} φ_{2}(x′′, σ(x′′))| ≤ \frac{7}{2}|\frac{∂φ_{2}}{∂x_{r_{2}}} (x′′, σ(x′′))|

and

|\frac{∂}{∂x_{r_{2}}} φ_{2}(x′′, σ(x′′))| ≥ \frac{1}{2}|\frac{∂φ_{2}}{∂x_{r_{2}}} (x′′, σ(x′′))| ≥ M_{2n−1}(1 + 2M_{2n−1}).

Consequently,

\frac{|\frac{∂}{∂x_{j}} φ_{2}(x′′, σ(x′′))|}{|\frac{∂}{∂x_{r_{2}}} φ_{2}(x′′, σ(x′′))|} ≤ 7, for any j ∈ \{1, …, n − 1\}.

In the same way, if φ_{1}(x′′, ρ(x′′)) ≢ +∞, we have almost everywhere on D

|\frac{∂}{∂x_{r_{1}}} φ_{1}(x′′, ρ(x′′))| ≥ M_{2n−1}(1 + 2M_{2n−1})

and

\frac{|\frac{∂}{∂x_{j}} φ_{1}(x′′, ρ(x′′))|}{|\frac{∂}{∂x_{r_{1}}} φ_{1}(x′′, ρ(x′′))|} ≤ 7, for any j ∈ \{1, …, n − 1\}.

By Lemma 6 (1), H admits an almost decomposition

(3.16)  H ≃ S_{1} ∪ ⋯ ∪ S_{k},

where every S_{ν} is either a semi-7-cell or a (\frac{1}{7}, 7)-cell in R^{n} after transposition (x_{r_{1}}, x_{n}).

Since G \ \overline{H} easily almost decomposes into a finite union of semi-M_{2n−1}-cells after transposition (x_{n−1}, x_{n}), (3.16) extends to a similar decomposition of G. Now, repeating the same argument for any (\frac{1}{7}, 7)-cell S_{ν} in the place of G with Lemma 6 (2) ends the proof in this case.

Case (3.11 ) Let φ_{i}^{−1} denotes the inversion of φ_{i} with respect to x_{n−1} (i ∈ {1, 2}).

Observe that if |\frac{∂φ_{i}}{∂x_{j}} | < 1 + 2M_{2n−1}, then

|\frac{∂φ_{i}^{−1}}{∂x_{j}} | = \begin{matrix}| & \frac{∂φ_{i}}{∂x_{j}} & | \\ | & \frac{∂φ_{i}}{∂x_{n−1}} & | \end{matrix} < 1 < 1 + 2M_{2n−1}

and, moreover,

|\frac{∂φ_{i}^{−1}}{∂x_{n}} | = \frac{1}{|\frac{∂φ_{i}}{∂x_{n−1}} |} < 1 < 1 + 2M_{2n−1}.

Hence,

♯\{j : |\frac{∂φ_{i}}{∂x_{j}} | < 1 + 2M_{2n−1}\} < ♯\{ν : |\frac{∂φ_{i}^{−1}}{∂x_{ν}} | < 1 + 2M_{2n−1}\} for i ∈ \{1, 2\}.

Then, after transposition (x_{n−1}, x_{n}), G is the following cell

G^{∗} = {(x′′, x_{n}, x_{n−1}) : x′′ ∈ Ω, φ_{1}(x′′, σ(x′′)) < x_{n} < φ_{2}(x′′, ρ(x′′)),
χ_{1}(x′′, x_{n}) < x_{n−1} < χ_{2}(x′′, x_{n})},

where

χ_{1}(x′′, x_{n}) = \{ \begin{matrix}σ(x′′) , & if φ_{1}(x′′, σ(x′′)) < x_{n} ≤ φ_{2}(x′′, σ(x′′)) \\ φ_{2}^{−1}(x′′, x_{n}) , & if φ_{2}(x′′, ρ(x′′)) < x_{n} < φ_{2}(x′′, ρ(x′′)) \end{matrix}

and

χ_{2}(x′′, x_{n}) = \{ \begin{matrix}φ_{1}^{−1}(x′′, x_{n}) , & if φ_{1}(x′′, σ(x′′)) < x_{n} < φ_{1}(x′′, ρ(x′′)) \\ ρ(x′′) , & if φ_{1}(x′′, ρ(x′′)) ≤ x_{n} < φ_{2}(x′′, ρ(x′′)). \end{matrix}

Since ⟨G^{∗}⟩ > ⟨G⟩, the induction hypothesis gives the desired decomposition.

Case (3.12 ) Then φ_{1}(σ) ⊂ R and define

ψ(x′′, x_{n−1}) = φ_{1}(x′′, σ(x′′)) + (1 + 2M_{2n−1})(x_{n−1} − σ(x′′)),

for (x′′, x_{n−1}) ∈ Δ. Now G splits into two cells:

S_{1} = {(x′, x_{n}) : x′ ∈ Δ, φ_{1}(x′) < x_{n} < ψ(x′)}

and

S_{2} = {(x′, x_{n}) : x′ ∈ Δ, ψ(x′) < x_{n} < φ_{2}(x′)}.

Observe that

\frac{∂ψ}{∂x_{j}} = \frac{∂φ_{1}}{∂x_{j}} + [\frac{∂φ_{1}}{∂x_{n−1}} − (1 + 2M_{2n−1})]\frac{∂σ}{∂x_{j}} ,

for j ∈ {1, …, n − 2}, almost everywhere on Δ.

Hence, by (3.8), (3.12) and (3.9),

|\frac{∂ψ}{∂x_{j}} | ≤ |\frac{∂φ_{1}}{∂x_{r_{1}}} | + 2M_{2n−1}(1 + 2M_{2n−1}) ≤ \frac{3}{2}|\frac{∂φ_{1}}{∂x_{r_{1}}} |

and

|\frac{∂ψ}{∂x_{r_{1}}} | ≥ |\frac{∂φ_{1}}{∂x_{r_{1}}} | − 2M_{2n−1}(1 + 2M_{2n−1}) ≥ \frac{1}{2}|\frac{∂φ_{1}}{∂x_{r_{1}}} | ≥ 2M_{2n−1}(1 + 2M_{2n−1}).

Therefore,

\begin{matrix}| & \frac{∂ψ}{∂x_{j}} & | \\ | & \frac{∂ψ}{∂x_{r_{1}}} & | \end{matrix} ≤ 3,

for any j ∈ {1, …, n − 2}. Thus S_{1} satisfies the conditions (3.13)–(3.15) and the case (3.10) applies.

On the other hand, if j ∈ {1, …, n − 2} and

|\frac{∂φ_{1}}{∂x_{j}} | < 1 + 2M_{2n−1},

then

|\frac{∂ψ^{−1}}{∂x_{j}} | = \begin{matrix}| & \frac{∂ψ}{∂x_{j}} & | \\ | & \frac{∂ψ}{∂x_{n−1}} & | \end{matrix} ≤ \frac{|\frac{∂φ_{1}}{∂x_{j}} | + 2M_{2n−1}(1 + 2M_{2n−1})}{1 + 2M_{2n−1}} < 1 + 2M_{2n−1};

hence,

♯\{j : |\frac{∂φ_{1}}{∂x_{j}} | < 1 + 2M_{2n−1}\} ≤ ♯\{ν : |\frac{∂ψ^{−1}}{∂x_{ν}} | < 1 + 2M_{2n−1}\},

while

♯\{j : |\frac{∂φ_{2}}{∂x_{j}} | < 1 + 2M_{2n−1}\} < ♯\{ν : |\frac{∂φ_{2}^{−1}}{∂x_{ν}} | < 1 + 2M_{2n−1}\}

and we finish by the induction hypothesis as in Case (3.11).

4. Theorem 1_{n} for a semi-M-cell.

Proposition 4. Any semi-M-cell G in R^{n} (where M > 0) admits an almost decomposition

(4.1)  G ≃ S_{1} ∪ ⋯ ∪ S_{k},

where every S_{ν} is an M′-cell after a permutation of coordinates and M′ ≥ 1 is a constant depending only on M and n.

Proof. One can assume that G is in the form (3.1), where φ_{i} : Δ ⟶ R (i = 1, 2) are continuous and

(4.2)  |\frac{∂φ_{1}}{∂x_{j}} | < M almost everywhere on Δ, for j ∈ \{1, …, n − 1\}.

Indeed, in the case φ_{1} ≡ −∞ or φ_{1} ≡ +∞ reduces to the above by assuming first that Δ is an M_{2n−1}-disc and applying next transposition (x_{n−1}, x_{n}).

The proof will be by descending induction on the number

[G] = ♯\{j : |\frac{∂φ_{2}}{∂x_{j}} | ≤ M_{2n−1} almost everywhere on Δ\}.

If [G] = n − 1, G is a max(M, M_{2n−1})-cell, so assume that [G] < n − 1. Notice that if Δ̃ ⊂ Δ, then for G̃ = G ∩ (Δ̃ × R), [G̃] ≥ [G].

Fix any L > max(M, M_{2n−1}) and any M^{∗} > M + (L + M)M_{2n−1}. Dividing Δ, one can assume that every φ_{i} is C^{1} on Δ and

(4.3)  for each j ∈ \{1, …, n − 1\}, sgn\frac{∂φ_{i}}{∂x_{j}} = const;

(4.4) for each j ∈ \{1, …, n − 1\}, |\frac{∂φ_{2}}{∂x_{j}} | > L on Δ or |\frac{∂φ_{2}}{∂x_{j}} | ≤ L on Δ

and

(4.5)  there exists r ∈ \{1, …, n − 1\} such that |\frac{∂φ_{2}}{∂x_{r}} | ≥ |\frac{∂φ_{2}}{∂x_{j}} |

for each j ∈ \{1, …, n − 1\}, and either |\frac{∂φ_{2}}{∂x_{r}} | ≥ M^{∗} or |\frac{∂φ_{2}}{∂x_{r}} | ≤ M^{∗} on Δ.

Clearly, one can assume that

(4.6)  |\frac{∂φ_{2}}{∂x_{r}} | ≥ M^{∗}  on Δ.

Finally, by Theorem 2_{n−1} and Lemma 2, one can assume that

Δ = {(x′′, x_{n−1}) : x′′ ∈ Ω, σ(x′′) < x_{n−1} < ρ(x′′)}

is an M_{2n−1}-disc in R^{n−1} and every φ_{i} admits a continuous extension

φ_{i} : Δ ∪ σ ∪ ρ ⟶ \overline{R}

such that φ_{i}(σ) ⊂ R or φ_{i}(σ) = {−∞}, or φ_{i}(σ) = {+∞}, and the same for ρ. Because of (4.2), φ_{1} : Δ ∪ σ ∪ ρ ⟶ R.

Case I :  |\frac{∂φ_{2}}{∂x_{n−1}} | > L on Δ.

Assume that \frac{∂φ_{2}}{∂x_{n−1}} > L; the case \frac{∂φ_{2}}{∂x_{n−1}} < −L will follow by a modification.

Consider the following function

(4.7)  ψ(x′′, x_{n−1}) = φ_{1}(x′′, σ(x′′)) + L(x_{n−1} − σ(x′′)),

for (x′′, x′) ∈ Δ.

Then φ_{1} < ψ < φ_{2} and G ≃ S_{1} ∪ S_{2}, where

S_{1} = {(x′, x_{n}) : x′ ∈ Δ, φ_{1}(x′) < x_{n} < ψ(x′)}

is an M^{∗}-cell and

S_{2} = {(x′, x_{n}) : x′ ∈ Δ, ψ(x′) < x_{n} < φ_{2}(x′)}

can be interpreted after transposition (x_{n−1}, x_{n}) as

S_{2} = {(x′′, x_{n−1}, x_{n}) : x′′ ∈ Ω, φ_{1}(x′′, σ(x′′)) < x_{n} < φ_{2}(x′′, ρ(x′′)),
θ_{2}(x′′, x_{n}) < x_{n−1} < θ_{1}(x′′, x_{n})},

where

θ_{2}(x′′, x_{n}) = \{ \begin{matrix}σ(x′′) , & if φ_{1}(x′′, σ(x′′)) < x_{n} ≤ φ_{2}(x′′, σ(x′′)) \\ φ_{2}^{−1}(x′′, x_{n}) , & if φ_{2}(x′′, σ(x′′)) < x_{n} < φ_{2}(x′′, ρ(x′′)) \end{matrix}

and

θ_{1}(x′′, x_{n}) = \{ \begin{matrix}ψ^{−1}(x′′, x_{n}) , & if φ_{1}(x′′, σ(x′′)) < x_{n} ≤ ψ(x′′, ρ(x′′)) \\ ρ(x′′) , & if ψ(x′′, ρ(x′′)) < x_{n} < φ_{2}(x′′, ρ(x′′)), \end{matrix}

and where φ_{2}^{−1} (and ψ^{−1}) stands for the inversion of φ_{2} (and ψ) with respect to x_{n−1}. Now, if j ∈ {1, …, n − 2} and

|\frac{∂φ_{2}}{∂x_{j}} | ≤ M_{2n−1},

then

|\frac{∂φ_{2}^{−1}}{∂x_{j}} | = \begin{matrix}| & \frac{∂φ_{2}}{∂x_{j}} & | \\ | & \frac{∂φ_{2}}{∂x_{n−1}} & | \end{matrix} < |\frac{∂φ_{2}}{∂x_{j}} | ≤ M_{2n−1}

and, moreover,

|\frac{∂φ_{2}^{−1}}{∂x_{n}} | = \frac{1}{|\frac{∂φ_{2}}{∂x_{n−1}} |} < \frac{1}{L} < M_{2n−1}.

Hence, [S_{2}] > [G] and the induction hypothesis ends the proof in this case.

Case II :  |\frac{∂φ_{2}}{∂x_{n−1}} | ≤ L on Δ.

By (4.6) and (4.3), one can assume without any loss of generality that

\frac{∂φ_{2}}{∂x_{r}} ≥ M^{∗} , \frac{∂φ_{2}}{∂x_{n−1}} > 0 and \frac{∂φ_{1}}{∂x_{n−1}} > 0 ;

other possibilities will follow by simple modifications.

Since M^{∗} > L, r ∈ {1, …, n − 2}. By Proposition 2, we have almost everywhere on Δ:

\frac{∂}{∂x_{r}} φ_{2}(x′′, σ(x′′)) = |\frac{∂φ_{2}}{∂x_{r}} (x′′, σ(x′′)) + \frac{∂φ_{2}}{∂x_{n−1}} (x′′, σ(x′′))\frac{∂σ}{∂x_{r}} (x′′)| ≥

M^{∗} − LM_{2n−1},

while

|\frac{∂}{∂x_{r}} φ_{1}(x′′, σ(x′′))| ≤ M + MM_{2n−1} and |\frac{∂}{∂x_{r}} φ_{1}(x′′, ρ(x′′))| ≤ M + MM_{2n−1}.

Thus, by Lemma 5,

φ_{2}(x′′, σ(x′′)) > φ_{1}(x′′, ρ(x′′)) on Ω.

Hence,

G ≃ S_{1} ∪ S_{2} ∪ S_{3},

where

S_{1} = {(x′′, x_{n−1}, x_{n}) : (x′′, x_{n−1}) ∈ Δ, φ_{1}(x′′, x_{n−1}) < x_{n} < φ_{1}(x′′, ρ(x′′))},
S_{2} = {(x′′, x_{n−1}, x_{n}) : x′′ ∈ Ω, φ_{1}(x′′, ρ(x′′)) < x_{n} < φ_{2}(x′′, σ(x′′)),
σ(x′′) < x_{n−1} < ρ(x′′)}

and

S_{3} = {(x′′, x_{n−1}, x_{n}) : (x′′, x_{n−1}) ∈ Δ, φ_{2}(x′′, ρ(x′′)) < x_{n} < φ_{2}(x′′, x_{n−1})}.

S_{1} is an M^{∗}-cell, while S_{2} is an M_{2n−1}-cell after transposition (x_{n−1}, x_{n}). We will investigate S_{3}. Put

Δ̃ = {(x′′, x_{n}) : x′′ ∈ Ω, φ_{2}(x′′, σ(x′′)) < x_{n} < φ_{2}(x′′, ρ(x′′))}.

16 WIESŁAW PAWŁUCKI

Now,

S_{3} = {(x′′, x_{n−1}, x_{n}) : (x′′, x_{n}) ∈ Δ̃, φ_{2}^{−1}(x′′, x_{n}) < x_{n−1} < ρ(x′′)},

where φ_{2}^{−1} stands for the inversion of φ_{2} with respect to x_{n−1}.

We will use Lemma 3 to get a desired decomposition of S_{3}. Observe first that

\frac{∂φ_{2}^{−1}}{∂x_{r}} = \frac{\frac{∂φ_{2}}{∂x_{r}}}{\frac{∂φ_{2}}{∂x_{n−1}}} ≥ \frac{\frac{∂φ_{2}}{∂x_{r}}}{L} ≥ \frac{M^{∗}}{L} > \frac{M + (L + M)M_{2n−1}}{L} > M_{2n−1} ≥ |\frac{∂ρ}{∂x_{r}} |

and

\begin{matrix}| & \frac{∂φ_{2}^{−1}}{∂x_{j}} & | \\ | & \frac{∂φ_{2}^{−1}}{∂x_{r}} & | \end{matrix} = \begin{matrix}| & \frac{∂φ_{2}}{∂x_{j}} & | \\ | & \frac{∂φ_{2}}{∂x_{r}} & | \end{matrix} ≤ 1 , for j ∈ \{1, …, n − 2\},

and

\begin{matrix}| & \frac{∂φ_{2}^{−1}}{∂x_{n}} & | \\ | & \frac{∂φ_{2}^{−1}}{∂x_{r}} & | \end{matrix} = \frac{1}{|\frac{∂φ_{2}^{−1}}{∂x_{r}} |} ≤ \frac{1}{M^{∗}} < 1.

Now it suffices to check that Δ has an almost decomposition into N-cells with respect to the variable x_{r}, where a constant N depends only on M, L, M^{∗} and M_{2n−1}. We will check this using Lemma 6 (2).

We have almost everywhere on Ω:

\frac{∂}{∂x_{r}} φ_{2}(x′′, σ(x′′)) ≥ \frac{∂φ_{2}}{∂x_{r}} (x′′, σ(x′′))(1 − \frac{LM_{2n−1}}{M^{∗}} ) ≥ M^{∗} − LM_{2n−1}

and

\frac{|\frac{∂}{∂x_{j}} φ_{2}(x′′, σ(x′′))|}{|\frac{∂}{∂x_{r}} φ_{2}(x′′, σ(x′′))|} ≤ \frac{|\frac{∂φ_{2}}{∂x_{j}} (x′′, σ(x′′)) + \frac{∂φ_{2}}{∂x_{n−1}} (x′′, σ(x′′))\frac{∂σ}{∂x_{j}} (x′′)|}{|\frac{∂φ_{2}}{∂x_{r}} (x′′, σ(x′′))|\frac{M(1 + M_{2n−1})}{M^{∗}}} ≤ \frac{M^{∗}}{M} .

The same is true for ρ in the place of σ. Moreover, by the assumption of Case II,

|φ_{2}(x′′, σ(x′′)) − φ_{2}(x′′, ρ(x′′))| ≤ |σ(x′′) − ρ(x′′)| on Ω.

Hence,

lim_{x′′→a′′}[φ_{2}(x′′, σ(x′′)) − φ_{2}(x′′, ρ(x′′))] = 0,

for any a′′ ∈ ∂Ω, so the assumptions of Lemma 6 (2) are satisfied.

5. Proof of Theorem 2_{n} for any M-cell.

Let

G = {(x′, x_{n}) : x′ ∈ Δ , φ_{1}(x′) < x_{n} < φ_{2}(x′)}

be any M-cell, where M ∈ R, M ≥ 1. Observe that all possible cases reduce to the case φ_{i} : Δ ⟶ R (i ∈ {1, 2}). Indeed, suppose for example that φ_{1} : Δ ⟶ R and φ_{2} ≡ +∞. Then one can assume first that φ_{1} is C^{1} on Δ and, for each j ∈ {1, …, n − 1},

sgn\frac{∂φ_{1}}{∂x_{j}} = const on Δ,

and next that

Δ = {(x′′, x_{n−1}) : x′′ ∈ Ω, σ(x′′) < x_{n−1} < ρ(x′′)}

is an M_{2n−1}-disc in R^{n−1} such that φ_{1} has a continuous extension

φ_{1} : Δ ∪ σ ∪ ρ ⟶ R.

Then, assuming that \frac{∂φ_{1}}{∂x_{n−1}} > 0,

G ≃ S_{1} ∪ S_{2},

where

S_{1} = {(x′′, x_{n−1}, x_{n}) : (x′′, x_{n−1}) ∈ Δ, φ_{1}(x′, x_{n−1}) < x_{n} < φ_{1}(x′′, ρ(x′′))}

is an M(1 + M_{2n−1})-cell, while

S_{2} = {(x′′, x_{n−1}, x_{n}) : x′′ ∈ Ω, φ_{1}(x′′, ρ(x′′)) < x_{n}, σ(x′′) < x_{n−1} < ρ(x′′)}

is an M_{2n−1}-cell after transposition (x_{n−1}, x_{n}).

Consequently, assume that φ_{i} : Δ ⟶ R (i ∈ {1, 2}) and that they are C^{1}. By Theorem 3_{n−1}, one can assume that Δ is a regular M_{3n−1}-cell and then, by Proposition 1, that every φ_{i} has a continuous extension

φ_{i} : \overline{Δ} ⟶ R (i ∈ \{1, 2\}).

Now, still keeping the last property, one can assume that

Δ = {(x′′, x_{n−1}) : x′′ ∈ Ω, σ(x′′) < x_{n−1} < ρ(x′′)}

is an M_{2n−1}-disc. Put

λ_{1}(x′′, x_{n−1}) = φ_{1}(x′′, σ(x′′)) + 2M(x_{n−1} − σ(x′′)),
λ_{2}(x′′, x_{n−1}) = φ_{1}(x′′, ρ(x′′)) − 2M(x_{n−1} − ρ(x′′)),
λ_{3}(x′′, x_{n−1}) = φ_{2}(x′′, ρ(x′′)) + 2M(x_{n−1} − ρ(x′′)),

18 WIESŁAW PAWŁUCKI

and

λ_{4}(x′′, x_{n−1}) = φ_{2}(x′′, σ(x′′)) − 2M(x_{n−1} − σ(x′′)),

for any (x′′, x_{n−1}) ∈ Ω×R. Every λ_{i} has a continuous extension to \overline{Ω}×R and is an M(1 + 3M_{2n−1})-function. Its inversion λ_{i}^{−1} with respect to x_{n−1} has a continuous extension to \overline{Ω} × R as well and is a \frac{1}{2}(1 + 3M_{2n−1})-function.

For any subset I ⊂ {1, 2, 3, 4}, put S_{I} = {(x′, x_{n}) ∈ G : x_{n} < λ_{i}(x′), if i ∈ I and λ_{i}(x′) < x_{n}, if i ∉ I}.

Then

G ≃ ⋃_{I} S_{I}.

It suffices to show that every S_{I} is an M(1 + 3M_{2n−1})-disc after perhaps transposition (x_{n−1}, x_{n}).

Fix any I ⊂ {1, 2, 3, 4}.

If {1, 2} ⊂ I, then

S_{I} = {(x′, x_{n}) ∈ Δ × R : φ_{1}(x′) < x_{n} < φ_{2}(x′), x_{n} < λ_{i}(x′), if i ∈ I,

λ_{i}(x′) < x_{n}, if i ∉ I},

and λ_{1} = φ_{1} on σ, while λ_{2} = φ_{1} on ρ and Lemma 4 applies.

Similarly, when {3, 4} ∩ I = ∅.

If {1, 2} ⊄ I and {3, 4} ∩ I ≠ ∅, we have 1 ∉ I and 3 ∈ I or 1 ∉ I and 4 ∈ I (or, similarly, 2 ∉ I and 3 ∈ I or 2 ∉ I and 4 ∈ I).

Suppose first that 1 ∉ I and 3 ∈ I. Then

(5.1) S_{I} = {(x′′, x_{n−1}, x_{n}) : x′′ ∈ Ω,  φ_{1}(x′′, σ(x′′)) < x_{n} < φ_{2}(x′′, ρ(x′′)),

σ(x′′) < x_{n−1} < ρ(x′′), x_{n−1} < λ_{i}^{−1}(x′′, x_{n}) if i ∈ Ĩ, λ_{i}^{−1}(x′′, x_{n}) < x_{n−1} if i ∉ Ĩ}, where Ĩ ⊂ {1, 2, 3, 4} is defined by the formula:

i ∈ Ĩ if and only if i ∈ I and i is even or i ∉ I and i is odd.

Since

λ_{1}^{−1}(x′′, φ_{1}(x′′, σ(x′′)) = σ(x′′)

and

λ_{3}^{−1}(x′′, φ_{2}(x′′, ρ(x′′)) = ρ(x′′),

for each x′′ ∈ Ω and

σ(x′′) = ρ(x′′),

for each x′′ ∈ ∂Ω, we are done by Lemma 4.

Let now 1 ∉ I and 4 ∈ I. Then (5.1) holds and since

λ_{1}^{−1}(x′′, φ_{1}(x′′, σ(x′′)) = σ(x′′), λ_{4}^{−1}(x′′, φ_{2}(x′′, σ(x′′)) = σ(x′′), for each x′′ ∈ Ω and σ(x′′) = ρ(x′′), for each x′′ ∈ ∂Ω, we are again done due to Lemma 4.

References

[vdD] L. van den Dries, Tame Topology and O-minimal Structures, Cambridge University Press, 1998.

[K] K. Kurdyka, On a subanalytic stratification satisfying a Whitney property with exponent 1, Proc. Conference Real Algebraic Geometry – Rennes 1991, Springer LNM 1524, pp. 316– 322.

[P] A. Parusiński, Lipschitz stratification of subanalytic sets, Ann. Scient. Ec. Norm. Sup. 27 (1994), 661–696.