MICHAEL’S THEOREM FOR LIPSCHITZ

CELLS IN O-MINIMAL STRUCTURES

Małgorzata Czapla and Wiesław Pawłucki

Instytut Matematyki Uniwersytetu Jagiellońskiego, ul. Prof. St. Łojasiewicza 6, 30-348 Kraków, Poland E-mail addresses: Malgorzata.Czapla@im.uj.edu.pl, Wieslaw.Pawlucki@im.uj.edu.pl

Abstract. A version of Michael’s theorem for multivalued mappings definable in o-minimal structures with M-Lipschitz cell values (M a constant) is proven.

1. Introduction. Assume that R is any real closed field and an expansion of R to some o-minimal structure is given. Throughout the paper we will be talking about definable sets and mappings referring to this o-minimal structure. (For fundamental definitions and results on o-minimal structures the reader is referred to [vdD] or [C].) In this article we adopt the following definition of a closed cell.

A subset S of R^{m} (m ∈ ℤ, m > 0) will be called a closed (respectively, closed M-Lipschitz) cell in R^{m}, where M ∈ R, M > 0, if

(i) S is a closed interval [α, β] (α, β ∈ R, α ⩽ β), or S = [α, +∞), or S = (−∞, α] (α ∈ R), or S = R, when m = 1 and

(ii) S = [f_{1}, f_{2}] := {(y′, y_{m}) : y′ ∈ S′, f_{1}(y′) ⩽ y_{m} ⩽ f_{2}(y′)}, where y′ = (y_{1}, …, y_{m−1}), S′ is a closed (respectively, closed M-Lipschitz) cell in R^{m−1}, f_{i} : S′ ⟶ R (i = 1, 2) are continuous (respectively, M-Lipschitz) definable functions such that f_{1}(y′) ⩽ f_{2}(y′), for each y′ ∈ S′, or S = [f, +∞) = {(y′, y_{m}) : y′ ∈ S′, y_{m} ⩾ f(y′)}, or S = (−∞, f] = {(y′, y_{m}) : y′ ∈ S′, y_{m} ⩽ f(y′)}, or S = S′ × R, where S′ is as before and f : S′ ⟶ R is continuous (respectively, M-Lipschitz), when m > 1.

Let F : A ⇉ R^{m} be a multivalued mapping defined on a subset A of R^{n}; i.e. a mapping which assigns to each point x ∈ A a nonempty subset F (x) of R^{m}. F can be identified with its graph; i.e. a subset of R^{n} × R^{m}. If this subset is definable we will call F definable. F is called lower semicontinuous if for each a ∈ A and each u ∈ F (a) and any neighborhood U of u, there exists a neighborhood V of a such that U ∩ F (x) ≠ ∅, for each x ∈ V.

The aim of the present article is the following theorem.

2010 Mathematics Subject Classification: Primary 14P10. Secondary 54C60, 54C65, 32B20, 49J53.

Key words and phrases: Michael’s theorem, Lipschitz cell, o-minimal structure.

Theorem 1. Let F : A ⇉ R^{m} be a definable multivalued, lower semicontinuous mapping defined on a definable subset A of R^{n} such that every value F (x) is a closed M-Lipschitz cell in R^{m}, where a constant M > 0 is independent of x ∈ A. Then F admits a continuous definable selection φ : A ⟶ R^{m}.

The following generalization of Theorem 1 is immediate.

Corollary 1. Let F : A ⇉ R^{m} be a definable multivalued, lower semicontinuous mapping defined on a definable subset A of R^{n}. If there is a continuous definable mapping Φ : A ⟶ Aut(R^{m}) with values in the space of linear automorphisms^{1} of R^{m} such that Φ(x)(F (x)) is a closed M-Lipschitz cell in R^{m}, then F admits a continuous definable selection φ : A ⟶ R^{m}.

Applying Theorem 1 to semilinear sets (see Remark 3 below) and taking into account that every closed semilinear cell is Lipschitz and for every semilinear family of semilinear cells they are M-Lipschitz with common M [vdD, Chapter 1, (7.4)], we obtain the following application generalizing [AT, Theorem 4.10]

Corollary 2. Let F : A ⇉ R^{m} be a semilinear multivalued, lower semicontinuous mapping defined on a semilinear bounded subset A of R^{n} such that every value F (x) is a closed semilinear cell in R^{m}. Then F admits a continuous semilinear selection φ : A ⟶ R^{m}.

For other results on multivalued mappings in connection with o-minimal geometry we refer the reader to [AT1], [AT2] and [DP].

2. Proof of Theorem 1.

The proof will be by induction on m. Consider first the case m = 1. Then F (x) = {t ∈ R : f(x) ⩽ t ⩽ g(x)}, for each x ∈ A, where f : A ⟶ R ∪ {−∞} and g : A ⟶ R ∪ {+∞} are definable functions.^{2} It is easy to check that F is lower semicontinuous if and only if g is lower semicontinuous and f is upper semicontinuous. Therefore, the problem reduces to the following.

Proposition 1. Let f : A ⟶ R∪{−∞} and g : A ⟶ R∪{+∞} be two definable functions such that f(x) ⩽ g(x), for each x ∈ A, and f is upper semicontinuous while g is lower semicontinuous. Then there exists a definable continuous function φ : A ⟶ R such that f ⩽ φ ⩽ g.

To prove Proposition 1, which is a definable version of the Katětov-Tong Insertion Theorem, we need the following definable version of the Tietze Theorem.

Theorem 2 (Definable Tietze’s Theorem). Let X and Y be two definable subsets of R^{n} such that Y is closed in X. Then every definable continuous function ψ : Y ⟶ R has a continuous definable extension Ψ : X ⟶ R.

For a proof of Theorem 2 see [vdD, Chapter 8, (3.10)] (compare also [AF, Lemma 6.6]).

Remark 1. According to [AT2, Theorem 3.3] Theorem 2 holds true in the semilinear o-minimal structure, provided that Y is bounded.

1The space Aut(R^{m}) is naturally identified with a subset of R^{m^{2}}.

2This means that f|f^{−1}(R) and g|g^{−1}(R) are definable.

Proof of Proposition 1. We use induction on d := dim A. The case d = 0 is trivial. Assume that d > 0. Let

B := {a ∈ A : f and g are both continuous in a neighborhood of a in A}. Then B is definable, open and dense subset of A. Hence A\B is definable closed in A and dim(A \ B) < d. By induction hypothesis there exists a definable continuous function ψ : A \ B ⟶ R such that for each x ∈ A \ B, f(x) ⩽ ψ(x) ⩽ g(x). By the Definable Tietze Theorem there exists a definable continuous extension Ψ : A ⟶ R of ψ. Now put φ(x) := min(max(Ψ(x), f(x)), g(x)), for each x ∈ A. It is clear that f ⩽ φ ⩽ g. Continuity of φ on B is obvious, since Ψ, f and g are continuous on B. We are checking continuity at any a ∈ A \ B. Then φ(a) = ψ(a) ∈ [f(a), g(a)]. Fix any ε > 0. There exists a neighborhood V of a in A such that ψ(a) + ε > f(x), ψ(a) − ε < g(x), ψ(a) + ε > Ψ(x) and ψ(a) − ε < Ψ(x) for each x ∈ V. Then φ(x) − ε = ψ(a) − ε < Ψ(x) ⩽ max(Ψ(x), f(x)) < ψ(a) + ε = φ(a) + ε and φ(a) − ε < g(x). Hence φ(a) − ε < φ(x) = min(max(Ψ(x), f(x)), g(x)) < φ(a) + ε.

Remark 2. Since Theorem 2 holds true for the o-minimal structure of semilinear sets under the assumption that X semilinear is bounded (see Remark 1), Proposition 1 holds true in this case too.

Assume now that m > 1 and our theorem is true for m − 1. To make the induction hypothesis work we prove the following.

Proposition 2. Under the assumptions of Theorem 1, let

π : R^{m} ∋ y = (y_{1}, …, y_{m}) ⟼ y′ = (y_{1}, …, y_{m−1}) ∈ R^{m−1}

be the natural projection. Let π ∘ F : A ⇉ R^{m−1} denote the composition defined by the formula (π ∘ F)(x) = π(F (x)).

Then F treated as a multi-valued mapping F : π ∘ F ⇉ R is lower semicontinuous.

Proof of Proposition 2. Put for each x ∈ A

F (x) = {(y′, y_{m}) : y′ ∈ π(F (x)), y_{m} ∈ R, f_{x}(y′) ⩽ y_{m} ⩽ g_{x}(y′)}.

Fix any (a, b′) ∈ π∘F, u ∈ F (a, b′) = {y_{m} ∈ R : f_{a}(b′) ⩽ y_{m} ⩽ g_{a}(b′)} and any open interval U_{ε} := (u − ε, u + ε). Let W be the open ball \{y′ ∈ R^{m−1} : |y′ − b′| < \frac{ε}{4M} \}, where | .| is defined by | y′| = |(y_{1}, …, y_{m−1})| = max_{j} | y_{j}|. By lower semi-continuity of F there exists a neighborhood V of a in A such that F (x) ∩ (W × U\frac{ε}{2} ) ≠ ∅, whenever x ∈ V.

Let now (x, y′) ∈ (π∘F)∩(V ×W). There exists (z′, v) ∈ F (x)∩(W ×U\frac{ε}{2} ). Then y′ ∈ π(F (x)), z′ ∈ π(F (x)); hence |y′ − z′| < \frac{ε}{2M} and f_{x}(z′) ⩽ v ⩽ g_{x}(z′). Thus, if f_{x} ≢ −∞, |f_{x}(y′)−f_{x}(z′)| ⩽ M|y′−z′| < \frac{1}{2}ε. Hence f_{x}(y′) ⩽ f_{x}(z′)+ \frac{1}{2}ε ⩽ v+ \frac{1}{2}ε < u+ε, also in the case when f_{x} ≡ −∞. Similarly, if g_{x} ≢ +∞, |g_{x}(y′)−g_{x}(z′)| < \frac{1}{2}ε and consequently g_{x}(y′) ⩾ g_{x}(z′)− \frac{1}{2}ε ⩾ v > u−ε. Finally, [f_{x}(y′), g_{x}(y′)]∩U_{ε} ≠ ∅, which ends the proof.

To finish the proof of Theorem 1, observe that the mapping π ∘ F is lower semicontinuous as a composition of a lower semicontinuous mapping with a continuous one, so by the induction hypothesis there exists a continuous definable selection φ′ for π ∘ F. By Proposition 2, F |φ′ : φ′ ⇉ R is lower semi-continuous; hence, by Proposition 1, it admits a continuous definable selection σ : φ′ ⟶ R, which gives a required selection φ = (φ′, σ ∘ (id_{A}, φ′)).

Remark 3. Proof of Proposition 2 holds true for the o-minimal structure of semilinear sets, so in view of Remark 2, the Theorem 1 holds true for the semilinear structure under the assumption that X semilinear is bounded.

3. A counterexample.

We are going to present an example of a semialgebraic mapping G : A ⇉ R^{2}, with A ⊂ R^{2}, which is not only lower semicontinuous, but even continuous with respect to the Hausdorff distance in the space of definable, closed, bounded and nonempty subsets, and which does not admit a continuous selection, although its values G(x_{1}, x_{2}) are M-Lipschitz cells but not with a constant M independent of (x_{1}, x_{2}). Let A = T_{1} ∪ T_{2}, where

T_{1} = {(x_{1}, x_{2}) : x_{1} ∈ [0, 1], −x_{1} ⩽ x_{2} ⩽ x_{1}}

and

T_{2} = {(x_{1}, x_{2}) : x_{1} ∈ [−1, 0], x_{1} ⩽ x_{2} ⩽ −x_{1}}.

We define G by the following

G(x_{1}, x_{2}) = \{ \begin{matrix}\{0\} × [0, 1] , & (x_{1}, x_{2}) = (0, 0), \\ \{(y, \frac{|y|}{|x_{1}|}) : −x_{1} + x_{2} ⩽ y ⩽ x_{1}\} , & x_{1} > 0, x_{2} ⩾ 0, \\ \{(y, \frac{|y|}{|x_{1}|}) : −x_{1} ⩽ y ⩽ x_{1} + x_{2}\} , & x_{1} > 0, x_{2} ⩽ 0, \\ \{(y, 1 − \frac{|y|}{|x_{1}|}) : x_{1} + x_{2} ⩽ y ⩽ −x_{1}\} , & x_{1} < 0, x_{2} ⩾ 0, \\ \{(y, 1 − \frac{|y|}{|x_{1}|}) : x_{1} ⩽ y ⩽ −x_{1} + x_{2}\} , & x_{1} < 0, x_{2} ⩽ 0. \end{matrix}

The graph of G is imagined by the following picture.

Suppose that the mapping G admits a continuous semialgebraic selection φ = (σ, ρ) : A ⟶ R^{2}. Then, for x_{1} > 0, σ(x_{1}, x_{1}) ⩾ 0 and σ(x_{1}, −x_{1}) ⩽ 0; hence, there exists ξ ∈ [−x_{1}, x_{1}] such that σ(x_{1}, ξ) = 0, so ρ(x_{1}, ξ) = \frac{|σ(x_{1},ξ)|}{|x_{1}|} = 0 and φ(x_{1}, ξ) = (0, 0). Consequently, by continuity, φ(0, 0) = (0, 0). Similarly, for any x_{1} < 0, there exists ξ ∈ [x_{1}, −x_{1}], such that φ(x_{1}, ξ) = (0, 1); hence φ(0, 0) = (0, 1), a contradiction.

Acknowledgement. We thank the referee for valuable comments.

References

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[AT2] M. Aschenbrenner; A. Thamrongthanyalak, Michael’s selection theorem in a semilinear context, Adv. Geom. 15 (2015), 293–313.

[AF] M. Aschenbrenner; A. Fischer, Definable versions of theorems by Kirszbraun and Helly, Proc. London Math. Soc. 102 (2011), 468–502.

[C] M. Coste, An Introduction to O-minimal Geometry, Dottorato di Ricerca in Matematica, Dipartimento di Matematica, Università di Pisa, Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000.

[DP] A. Daniilidis; J.C.H. Pang, Continuity and differentiability of set-valued maps revisited in the light of tame geometry, J. London Math. Soc. 83 (2011), 637–658.

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