APPROXIMATION OF O-MINIMAL MAPS SATISFYING A
LIPSCHITZ CONDITION
ANDREAS FISCHER
Abstract. Consider an o-minimal expansion of the real field. We show that
definable Lipschitz continuous maps can be definably fine approximated by
definable continuously differentiable Lipschitz maps whose Lipschitz constant
is close to that of the original map.
1. Introduction
Let n and k denote strictly positive natural numbers. We endow Rn and Rk with
the Euclidean norm k·k and the corresponding topology. Let U ⊂ Rn be an open
set, and let L > 0. Let f : U → Rk be a Lipschitz continuous map with Lipschitz
constant L; that is: for all u, v ∈ U , the function f satisfies the inequality
kf (u) − f (v)k ≤ L ku − vk .
From the geometrical point of view, one wishes to have a continuously differentiable
approximation map g of f which shares as many geometrical properties as possible
with the map f . Naturally, the main focus is set on the Lipschitz constant.
For example, if k = 1, we obtain by a well known application of integral convolution that f can be uniformly approximated by a smooth Lipschitz function g
of the same Lipschitz constant as f . This statement holds true for Lipschitz maps
from c0 -spaces Banach spaces satisfying some extra properties; see [5, 16, 18, 19].
For many geometric applications, the stronger concept of fine approximation is
required. In this case, the Lipschitz constant of the approximating function g is
slightly bigger; i.e. if f is an L-Lipschitz function and ε > 0, then one can choose g
to be sufficiently smooth and L + ε-Lipschitz. As long as real valued Lipschitz functions are considered, this result generalizes to functions on Riemannian manifolds
and Hilbert spaces; see for example [2, 3].
If maps are considered, then, there is a constant M ≥ 1, which depends on
the source space or the target space, such that every L-Lipschitz map can be finely
approximated by a smooth M L+ε-Lipschitz map. In case of the√target space being
Rk (still equipped with the Euclidean norm) we can take M = k. Notably, there
is uniform and fine approximation theorem for Lipschitz maps between separable
Banach spaces, if target or source space is a C(K) space. There, we can take
M = 1; see [18, 19]. However, a C(K) space is equipped with the supremum-norm.
Recall that Rn is always equipped with the Euclidean norm in the present paper!
The proofs given in the above cited papers make essential use of infinite partitions of unity and sometimes also of integration. These methods do not preserve
definability. Both the coarse control of the constant M and the transcendence of
Date: November 3, 2011.
2000 Mathematics Subject Classification. Primary 03C64, 14P99, 26B35 Secondary 26B05.
1
2
ANDREAS FISCHER
the known proofs prompt us to analyze fine approximation of Lipschitz maps for ominimal structures. (See [7, 8] for this, and basic definitions concerning o-minimal
structures, and [9, 10, 17, 21, 25] for some examples of o-minimal structures). More
precisely, we study the following question:
If f is any Lipschitz map which is definable in an o-minimal expansion R, does
f admit a definable close approximating Lipschitz map g which is continuously differentiable and whose Lipschitz constant is close to that of f ?
Questions of this nature are also of general interest as many but not all properties
known in real analysis and topology hold true in an o-minimal setting, even if the
underlying set is a general real closed field. However, we were not successful in
finding a reference indicating that fine approximation of Lipschitz maps between
Euclidean spaces almost preserving the Lipschitz constant is possible.
Therefore, let R be an o-minimal expansion of R. In the sequel, definable always
means definable with parameters in R.
The main result to be proved in the present paper and which implies an affirmative answer to the question above is the definably fine approximation of definable
locally Lipschitz continuous functions. Recall that a function f : U → Rk is called
locally Lipschitz continuous or just locally Lipschitz if for every u ∈ U there exists
an open ball Br (u) with radius r and center x, such that the restriction of f to
Br (u) is L-Lipschitz for some L > 0.
Let f : U → Rk be locally Lipschitz continuous and definable. The local Lipschitz
constant of f at a point u ∈ U is
kf (v) − f (w)k
| v, w ∈ Br (u) ∧ v 6= w .
lipf (u) = inf sup
r>0
kv − wk
Let ` ∈ N ∪ {∞, ω}. We use C ` as abbreviation for ` times continuously differentiable. For a function f we let Sing` (f ) denote its set of C ` -singular points.
Uniform approximation of locally Lipschitz functions has been studied in [6] and
[24]. However, we are interested in fine approximation. In the case of approximation
of locally Lipschitz maps, one is interested in the control of the local Lipschitz
constants. Thus, we shall prove the following theorem:
Theorem 1. Let U ⊂ Rn be definable and open, let f : U → R be a definable locally
Lipschitz function, and let e : U → (0, ∞) be a definable continuous function. Let
V ⊂ U be a definable open set containing the closure of Sing1 (f ) in U . Then for
every ε > 0 there is a definable C 1 function g : U → R such that
(a) |g − f | < e on U ,
(b) lipg (u) ≤ (1 + ε)lipf (u) + ε for all u ∈ U ,
(c) g = f on U \ V .
An important consequence of the previous theorem is the following definable
approximation of Lipschitz functions.
Theorem 2. Let L > 0. Let U ⊂ Rn be open, let f : U → R be a definable LLipschitz function, and let e : U → (0, ∞) be a definable continuous function. Let
APPROXIMATION OF O-MINIMAL MAPS SATISFYING A LIPSCHITZ CONDITION
3
V ⊂ U be a definable open set containing the closure of Sing1 (f ) in U . Then for
every rational number ε > 0 there is a definable C 1 function g : U → R such that
(a) g is L + ε-Lipschitz,
(b) |g − f | < e on U ,
(c) lipg (u) ≤ lipf (u) + ε for all u ∈ U ,
(d) g = f on U \ V .
If n = 1, then we can choose ε = 0 by the cost of dropping item (c), see [13,
Prop. 2]. It is worth noting that our proof works for general Archimedean real
closed fields in place of R.
A priori, item (a) and (b) do not imply a similar approximation statement for
maps. This requires item (c). Item (d) has turned out to be useful for applications;
see [15].
An (embedded) C 1 manifold is called definable C 1 manifold if its underlying set
is definable. Theorem 2 allows us to prove the answer to the above question in a
slightly more general form.
Theorem 3. Let L > 0, and let ε > 0. Let M ⊂ Rn and N ⊂ Rk be definable C 1
manifolds, let f : M → N be a definable L-Lipschitz map, and let e : M → (0, ∞)
be definable and continuous. Then there is a definable C 1 -smooth L + ε-Lipschitz
map g : M → N such that kg − f k < e on M .
The fine approximation evidently implies the uniform approximation of definable
Lipschitz maps. Nevertheless, for bounded definable Lipschitz maps, we obtain as
an application of Theorem 2 the uniform approximation of definable Lipschitz maps
preserving the Lipschitz constant.
Corollary 4. Let L > 0, ε > 0, and let f : U → Rk be a bounded definable LLipschitz map. Then there is a definable C 1 -smooth L-Lipschitz map g such that
|g − f | < ε on U .
The paper is organized as follows: In Section 2 we study subtle properties of
the C 1 singularities of definable locally Lipschitz functions which are crucial for the
proof of our approximation theorem. Section 3 is devoted to the proof of Theorem
1 and Section 4 for the proofs of the other Theorems stated in the introduction.
In the final Section 5 we briefly discuss further applications and variants of the
previous theorems and also address to the locally definable context in both the
senses of van den Dries/Miller and Shiota.
2. Partial Lipschitz derivatives
2.1. For 1 ≤ d ≤ n and x = (x1 , . . . , xn ) we let x0 = (x1 , . . . , xd ) and x00 =
(xd+1 , . . . , xn ).
For a set X we let cl(X) and fr(X) = cl(X) \ X denote its topological closure
and frontier, respectively.
Let k ∈ N ∪ {∞, ω}. A definable map f : A → Rd , where A is not necessarily
open, is called C k -smooth if there exists an open definable set U containing A and
a definable C k map g : U → Rd such that f = g on A.
4
ANDREAS FISCHER
2.2. The dimension of a definable set is the maximal integer d such that A contains
a definable set which is definably homeomorphic to Rd . This definition is welldefined; see [8, Ch. 4] for a detailed description of dimension. Note that every
nonempty definable set has dimension.
A definable subset A of a nonempty definable set B is called thick (in B) if
dim(B \ A) < dim(B).
2.3. Let U ⊂ Rn be definable and open. If f : U → R is a definable function, then
the set of points at which f is C 1 -smooth is definable and thick in U ; see [8].
Let I ⊂ R be some open interval, and let g : I → R be a locally Lipschitz
continuous function. Denote by I 0 (g) the set of points at which f is differentiable
(in the usual sense).
L
Let x ∈ I. The Lipschitz derivative ddxg (x) of g at x is defined by
dL g
1
(x) = sup {g 0 (t), t ∈ (x − ε, x + ε) ∩ I 0 } +
dx
2 ε>0
1
+ inf {g 0 (t), t ∈ (x − ε, x + ε) ∩ I 0 }
2 ε>0
Similar to the partial derivative and the gradient in the ordinary sense, we obtain
the partial Lipschitz derivative ∂ L f /∂xi with respect to the ith coordinate and the
Lipschitz gradient ∇L f . To simplify notation we write ∂iL f in place of ∂ L f /∂xi .
Remark 1.
(a) If g : I → R is definable and locally Lipschitz continuous, then I \ I 0 (g) is
a finite set.
(b) If f : U → R is definable, locally Lipschitz continuous and differentiable
in the usual sense, then the partial Lipschitz derivatives coincide with the
usual partial derivatives. This follows from the fact that unary definable
differentiable functions are continuously differentiable; cf. [14].
(c) If f is not definable anymore, but differentiable and locally Lipschitz continuous, then ∂f (x) and ∂ L (x) do not necessarily coincide anymore.
Example 1. Let zack : R → [0, 1] be defined


0,
zack(x) = 2x,


2 − 2x,
by
if x 6∈ [0, 1],
if 0 ≤ x ≤ 21 ,
if 12 < x ≤ 1.
Define g : R → R by
g(t) =
∞
X
zack 22n t − 2−n
n=1
Then f : R → R, defined by
x
Z
f (x) :=
g(t)dt,
0
is differentiable and 1-Lipschitz continuous. However,
∂1 f (0) = 0 6=
1
= ∂1L f (0).
2
.
APPROXIMATION OF O-MINIMAL MAPS SATISFYING A LIPSCHITZ CONDITION
5
Remark 2. Let f : U → R be a definable locally Lipschitz continuous function. By
the very definition of the local Lipschitz constant, we have the inequality
L
∇ f (u) ≤ lipf (u)
for every u ∈ U . Moreover, the function
u 7→ lipf (u)
is definable
2.4. We recall the well-known fact that a definable L-Lipschitz function f : U → R
can always be extended to a definable L-Lipschitz function f on Rn using the
McShane-Whitney operator which maps f to f : Rn → R defined by
f (v) = inf{L kv − uk + f (u) | u ∈ U }.
The next lemma names conditions to assume Lipschitz continuity for definable
differentiable functions with bounded derivative. Note that in general a continuously differentiable function of several variables with bounded derivative is not
Lipschitz continuous.
In the sequel, the symbol ∇ is used to denote the gradient operator.
Lemma 5. Let U ⊂ Rn be open and f : U → R be definable and L-Lipschitz
continuous. Let V ⊂ U be open and g : V → R be definable and continuously
differentiable such that its gradient is bounded by L, and such that
(
g(ξ), ξ ∈ V,
F (ξ) :=
f (ξ), ξ ∈ cl(U ) \ V
is continuous. Then F is L-Lipschitz continuous.
Proof. Let x, y ∈ Rn . Consider the function φ : [0, kx − yk] → R, defined by
y−x
φ(t) := f x + t
.
ky − xk
Then
L
∂1 φ(t) ≤ L
for all t. So φ is L-Lipschitz by the Mean-Value-Theorem. Hence,
|f (x) − f (y)| = |φ(0) − φ(ky − xk)| ≤ L kx − yk
as required.
2.5. Singularity sets of partial Lipschitz derivatives. The C 1 -singularity sets
of definable Lipschitz functions are of codimension at least 1. Conversely, every definable set of codimension at least one happens to be the C 1 -singularity set of a
definable Lipschitz function. For our construction, however, we need more information about the partial Lipschitz derivatives. By the use of a suitable stratification
we will be able to reduce our considerations to the case, where the singularity sets
are thick subsets of coordinate hyperplanes. There, the partial Lipschitz derivatives with respect to coordinates parallel to the hyperplane are continuous with
the exception of a definable subset of codimension at least 2. We prove this in the
following proposition.
Proposition 6. Let X ⊂ Rd be an open definable set, and let U ⊂ Rn be an open
definable neighbourhood of X × {0}n−d in Rn . Suppose that the definable function
f : U → R is locally Lipschitz continuous such that
6
ANDREAS FISCHER
(a) f restricted to X × {0}n−d is C 1 -smooth,
(b) f restricted to the complement of X × {0}n−d in U is C 1 -smooth.
Then the dimension of the set of points at which
∂iL f,
i = 1, . . . , d
is not continuous is less than or equal to d − 1.
Proof. Without loss of generality, we may assume that i = 1, otherwise we suitably
permute the coordinates. Moreover, by considering
(x0 , x00 ) 7→ f (x0 , x00 ) − f (x0 , 0)
in place of f , we may assume that f |X = 0.
Let
X̃ = X × {0}n−d .
We assume - for a contradiction - that ∂1L f is discontinuous on a subset of X̃ of
dimension d.
Then there is a constant M > 0 such that either
dim fr{∂1L f > M } ∩ X̃ = d
or
dim fr{∂1L f < −M } ∩ X̃ = d.
We will work with the first case since the second case is similar.
By cell decomposition, there is an open cell
S ⊂ {∂1L f > M }
with
dim fr(S) ∩ X̃ = d.
Let x be an element of the relative open kernel of fr(S) ∩ X̃ in X̃. Let B be an open
ball with center x whose closure is contained in U . So the local Lipschitz constants
of f restricted to B are bounded by the number
K := sup lipf (x).
x∈B
By the Curve-selection-lemma, there is a continuous definable curve
ϕ : (0, δ] → S ∩ B
with
lim ϕ(t) = x.
t&0
Denote by e1 the unit-vector of the first coordinate. The choice of x implies that
there is an ε > 0 such that
{ϕ(t) + se1 | s ∈ [0, ε] & t ∈ (0, δ)} ⊂ S.
Now, by the Mean-Value-Theorem, there is for every t an element ξt of the linesegment connecting ϕ(t) and ϕ(t) + εe1 such that
f (ϕ(t)) − f (ϕ(t) + εe1 )
= ∂1L f (ξt ).
ε
However, we can estimate the left side of the previous equation as follows:
f (ϕ(t)) − f (ϕ(t) + εe1 ) 2K kϕ(t) − xk
≤
;
ε
ε
APPROXIMATION OF O-MINIMAL MAPS SATISFYING A LIPSCHITZ CONDITION
7
and the right side tends to 0 as t → 0. But then
ξt 6∈ {∂1L f > M }
for small t. This is a contradiction.
3. Preparation of the proof of Theorem 2
3.1. A major tool of our proof is Λ1 -regular stratification. We briefly recall this
concept.
A Λ1 -regular map, cf. [12, Def. 1.3] or [20], is a definable C 1 map with bounded
first derivative. The set of Λ1 -regular maps from U to V is denoted by Λ1 (U, V ).
The symbols ±∞ are regarded as constant functions defined on arbitrary sets.
A Λ1 -regular standard cell in R is either a single point or an open interval.
Suppose we know the standard Λ1 -regular cells in Rn−1 , then a Λ1 -regular standard
cell S of Rn is either a single point, or a definable set S of either the form
S = Γ(h)
1
k
where h ∈ Λ (X, R ) is defined on some open Λ1 -regular standard cell X ⊂ Rn−k ;
or of the form
S = {(x, y) : x ∈ X, f (x) < y < g(x)}
for some open Λ1 -regular standard cell X ⊂ Rn−1 and functions f, g ∈ Λ1 (X, R) ∪
{±∞} such that f < g on X.
A set S ⊂ Rn is called a Λm -regular cell if there is an orthogonal linear automorphism of Rn mapping S to a standard Λm -regular cell.
Let δ > 0. A Λ1 -regular cell is called δ-flat if it is either an open set, or the
map h of one of its corresponding standard cells S = Γ(h) is additionally Lipschitzcontinuous with constant δ.
Our interest in Λ1 -regular cells is based on the following kind of stratification,
cf. [12, Thm. 1.4 & proof of Prop. 4.1].
Theorem 7. Let δ > 0. For every finite family of definable subsets A1 , . . . , Ak of
Rn , there exists a δ-flat Λ1 -stratification of Rn which is compatible with the sets
A1 , . . . , Ak . That is, there is a finite partition of Rn into subsets S1 , . . . , Sr , called
strata, such that
(a) each stratum is a δ-flat Λ1 -regular cell,
(b) for each stratum Si , the frontier fr(Si ) := cl(Si ) \ Si is the union of some
of the strata,
(c) each Aj is the union of some of the strata.
Remark 3. The previous Theorem holds true over general real closed field if and
only if δ is chosen to be greater than some positive rational number.
3.2. Firstly we prove a technical lemma which we will need for the proof of Theorem
2.
Lemma 8. Let δ > 0. Let X ⊂ Rd be a definable open set, and let V be any open
definable neighbourhood of X × {0}n−d in Rn . Then there is a definable C 1 -function
σ : X → [0, 1] which tends to 0 as x tends to the boundary of X or infinity such
that V contains the set
Wσ := {(x, y) ∈ X × Rn−d : kyk < σ(x)},
8
ANDREAS FISCHER
and
k∇σk < δ.
Proof. Let the semi-algebraic C ω -map φ : Rd → (−1, 1)d be given by
q
q
φ(x1 , . . . , xd ) = (x1 / 1 + x21 , . . . , xd / 1 + x2d ).
The gradient of this map is bounded, and the image φ(X) is bounded and open.
We select a definable C 1 function θ : Rd → R which vanishes outside of φ(X)
and which is strictly positive on φ(X). The support of θ is bounded, so its gradient
is bounded, too.
The zero-set of D : Rd → R, defined by
x 7→ dist((x, 0), Rd \ ψ(V 0 )),
is contained in the zero-set of θ. This allows us to apply the generalized Lojasiewicz
inequality, cf. [7, Theorem C14], to θ and D. Thus, we obtain a bijective definable
C 1 function ρ : R → R with ρ(0) = 0 such that
ρ ◦ θ(x) ≤ D(x)
d
for x ∈ R . So σ = kρ ◦ θ ◦ φ (where k is some sufficiently small positive real
number) is the desired function.
3.3. A useful function. Let 2 ≤ m ∈ N. For the smoothing process of Lipschitz
functions we will make use of the following function
ρm : [0, ∞) → [0, 1]
which is defined by
(1)

 m t1/m − t ,
m−1
ρm (t) := m − 1
1,
if 0 ≤ t ≤ 1,
otherwise.
We notice:
• ρ restricted
 to the interval (0, 1) is continuously differentiable,
 1 t1/m−1 − 1 , if 0 ≤ t ≤ 1,
0
m−1
• ρ (t) = m − 1
0,
otherwise.
Hence, we have the estimates:
1
• |ρ0 (t)| ≤
t1/m−1 , t > 0
m−1
• ρ(t) ≤ 2t1/m , t > 0.
3.4. Next we prove the approximation for the following special case:
Lemma 9. Let X ⊂ Rd be a definable open set, and let ε > 0. Let U ⊂ X × Rn−d .
Let e : U → (0, ∞) be a definable continuous function, and let V be an open
definable set between X × {0}n−d and U .
Let F : U → R be a definable locally Lipschitz continuous function whose restriction to U \ X × {0}n−d and X × {0}n−d is C 1 -smooth.
If for all i = 1, . . . , d the functions ∂iL F are continuous on U , then there is a
definable C 1 function G : U → R such that
(a) |G − F | < e on U ,
APPROXIMATION OF O-MINIMAL MAPS SATISFYING A LIPSCHITZ CONDITION
9
(b) lipg (u) ≤ (1 + ε)lipf (u) + ε for every u ∈ U ,
(c) G = F on U \ V .
Proof. Let σ : X → [0, 1] be a definable C 1 function from Lemma 8 with δ := ε/(2n)
such that Wσ is contained in the intersection of the sets
V,
\ ε
∂iL F (x0 , x00 ) − ∂iL F (x0 , 0) ≤ √
Y =
2 n
i≤d
and
0
00
(x , x ) ∈ X × R
Z=
n−d
e(x0 )
| (lipF (x , 0) + ε) kx k <
2 + 2ε
0
00
.
Choose an integer m > 1 such that
1
ε
≤ √ .
m−1
2 n
Let ρ := ρm , where ρm is the function from (1).
We claim that the following function G : U → R defined by
00 kx k
G(x0 , x00 ) := F (x0 , 0) + (F (x0 , x00 ) − F (x0 , 0))ρ
σ(x0 )
satisfies the properties we look for.
First, we note that the function
x0 7→ F (x0 , 0)
is continuously differentiable by assumption, and that G is C 1 -smooth for kx00 k > 0
by construction. Moreover, for (x0 , x00 ) with kx00 k small enough, we obtain the
estimate
00 00 1/m
(F (x0 , x00 ) − F (x0 , 0))ρ kx k ≤ (lipF (x0 , 0) + 1) kx00 k 2 kx k
σ(x0 ) σ(x0 )
1+1/m
= 2(lipF (x0 , 0) + 1)
So
kx00 k
σ(x0 )1/m
kx00 k
(F (x , x ) − F (x , 0))ρ
is o(kx00 k) as kx00 k → 0.
σ(x0 )
Therefore, the function G is differentiable on U in the usual sense.
Next we analyze the partial derivatives ∂i G restricted to W .
0
00
0
Case 1: 1 ≤ i ≤ d.
Then
0
00
∂i G(x , x )
kx00 k
+
−
ρ
+
σ(x0 )
|
{z
}
√
| {z }
|. . . | ≤ ε/2 n
≤1
00 kx k − kx00 k ∂i σ(x0 )
+
(F (x0 , x00 ) − F (x0 , 0))
ρ0
|
{z
}
σ(x0 )
σ 2 (x0 )
0
00
|. . . | ≤ (lipF (x , 0) + ε) kx k
=∂iL F (x0 , 0)
∂iL F (x0 , x00 )
∂iL F (x0 , 0)
10
ANDREAS FISCHER
The absolute value of the last summand can be estimated by
00 1/m−1 00
1
kx k
kx k |∂i σ(x0 )|
0
00
|. . . | ≤ (lipF (x , 0) + ε) kx k
m − 1 σ(x0 )
σ 2 (x0 )
1/m+1
(lipF (x0 , 0) + ε) kx00 k
=
|∂i σ(x0 )|
m−1
σ(x0 )
1/m+1
(lipF (x0 , 0) + ε)ε kx00 k
√
(2)
≤
σ(x0 )
2 n
0
(lipF (x , 0) + ε)ε
√
≤
2 n
Case 2: d + 1 ≤ i ≤ n.
Then
∂Gi (x0 , x00 ) = ∂iL F (x0 , x00 ) ρ
|
+
kx00 k
+
σ(x0 )
{z }
≤1
0
00
0
ρ
(F (x , x ) − F (x , 0))
{z
}
|
0
00
|. . . | ≤ (lipF (x , 0) + ε) kx k
0
kx00 k
σ(x0 )
xi
kx00 k σ(x0 )
The last summand can be estimated by
00 1/m−1
kx k
|xi |
1
m − 1 σ(x0 )
kx00 k σ(x0 )
00 1/m−1
ε
kx k
kxi k
≤ (lipF (x0 , 0) + ε) √
0
σ(x0 )
2 n σ(x )
1/m
(lipF (x0 , 0) + ε)ε kx00 k
√
≤
σ(x0 )
2 n
0
(lipF (x , 0) + ε)ε
√
≤
2 n
|. . . | = (lipF (x0 , 0) + ε) kx00 k
(3)
The inequalities (2) and (3) imply that the partial derivatives are continuous, so
G is C 1 -smooth.
Furthermore, we obtain the inequality
L L (lipF (x0 , 0) + ε)ε
∂i G ≤ ∂i F +
√
2 n
on W for all i = 1, . . . , n. Hence:
k∇Gk ≤ k∇F k + (lipF (x0 , 0) + ε)ε
on W \ X × {0}n−d . So
lipG (u) ≤ (1 + ε)lipF (u) + ε2
for all u ∈ U . This proves item (b). In connection with Lemma 5 this implies item
(a). By the choice of the set W ⊂ Z, item (c) is evident.
We are now able to proof Theorem 1
APPROXIMATION OF O-MINIMAL MAPS SATISFYING A LIPSCHITZ CONDITION
11
Proof of the Theorem 1. Let f : U → R be a definable locally Lipschitz continuous
function. Let V ⊂ U be a definable open neighbourhood of the closure of Sing1 (f )
in U . (Notice that Sing1 (f ) is definable.)
It is enough to prove the following statement by induction on d:
Let
dim(Sing1 (f )) < d.
Then, for every definable continuous e : U → (0, ∞), and for every number δ > 0,
there is a definable C 1 -smooth locally Lipschitz continuous function g : U → R such
that
(a) |f − g| ≤ e on U ,
(b) g = f on U \ V ,
(c) lipg (u) ≤ (1 + δ)lipf (u) + δ 2 .
In the case d = 0, nothing has to be proved.
We assume that the statement holds true for d − 1.
Let δ 0 be a positive rational number such that
(1 + δ 0 )3 < 1 + δ/3.
Note that δ 0 < δ/2, so (1 + δ 0 )δ 0 < δ.
Let S1 , . . . , Sk be a δ 0 -flat Λ1 -regular stratification of U which is compatible with
Sing1 (f ). Let S1 , . . . , Sk0 denote the cells of dimension d which are contained in
Sing1 (f ). Choose for every j = 1, . . . , k 0 a definable open neighbourhood Uj of Sj
such that
Uj ∩ Uj 0 = ∅ for j 6= j 0 .
Such neighbourhoods exist by the properties of a stratification. By C 1 cell decomposition, we may further assume that f restricted to Uj \ Sj is C 1 -smooth.
For every j = 1, . . . , k 0 we will change the values of f on Uj .
By Theorem 7, choose for every j = 1, . . . , k 0 a suitable linear orthogonal coordinate system such that we can write
Sj = (hj )Zj ,
where Zj ⊂ Rd is an open (Λ1 -regular) cell and hj : Zj → Rn−d a δ 0 -flat C 1 -map
which is δ 0 -Lipschitz continuous.
Consider the map
φj : Zj × Rn−d → Zj × Rn−d
defined by
φj (x0 , x00 ) = (x0 , x00 + hj (x)).
0
Then φj is a (1 + δ )-Lipschitz C 1 map whose inverse is also a (1 + δ 0 )-Lipschitz
continuous C 1 map. Set
Fj := f ◦ φj .
Then
φ−1 (Uj ) ∩ Sing1 (Fj ) ⊂ Zj × {0}n−d .
Set
Uj0 = φ−1 (Uj ∩ (Zj × Rn−d )).
12
ANDREAS FISCHER
Thus, for i = 1, . . . , d, the partial Lipschitz derivatives ∂iL Fj are continuous on a
definable thick subset Xj of Zj by Proposition 6.
Select for every j = 1, . . . , k 0 a definable open neighbourhood Vj of Xj such that
Vj ⊂ Uj0
and
(Vj \ frXj ) ⊂ Uj0 .
Apply Lemma 9 to Fj , Uj0 , Xj , Vj , δ 0 and e/2 in place of F , U , X, V , ε and e to
obtain a definable locally Lipschitz continuous function Gj : Uj0 → R such that
lipGj (u) ≤ (1 + δ 0 )lipFj (u) + δ 0
for all u ∈ Uj0 , and Gj = Fj outside of Vj
Let g̃ : U → R be defined by
(
0
Gj ◦ φ−1
j (u), if u ∈ ∪j φ(Uj ),
g̃(u) :=
f (u),
if u ∈ U \ ∪j Uj .
Then g̃ is C 1 smooth with the exception of a definable d − 2-dimensional subset
of cl(Sing1 (f )) such that
e
|g̃ − f | <
2
on U . The local Lipschitz constants can be estimated as follows:
lipg̃ (u) ≤ (1 + δ 0 )lipGj (φ−1
j (u))
0 0
≤ (1 + δ 0 )2 lipf ◦φj (φ−1
j (u)) + (1 + δ )δ
≤ (1 + δ 0 )3 lipf (u) + (1 + δ 0 )δ 0
Choose δ 00 > 0 so small that both
(1 + δ 00 )(1 + δ/3) < 1 + δ
and
(1 + δ 00 )(1 + δ 0 )δ 0 + δ 00 < δ.
By the induction hypothesis, there is a definable locally Lipschitz continuous C 1
function g : U → R such that
|g − g̃| < e/2
on U , such that
lipg (u) < (1 + δ 00 )lipg̃ (u) + δ 00
for all u. Hence, the choice of δ 00 implies that
lipg (u) < (1 + δ)lipf (u) + δ.
Hence g is the desired function,
APPROXIMATION OF O-MINIMAL MAPS SATISFYING A LIPSCHITZ CONDITION
13
4. Proof of Theorem 2 and 3
Proof of Theorem 2. Let L > 0 be the Lipschitz constant of f : U → R. Choose
δ > 0 so small that
Lδ + δ < ε.
By Theorem 1, there is a definable C 1 function g such that |g − f | < e on U , g = f
on U \ V and
lipg (u) < (1 + δ)lipf (u) + δ.
Hence
lipg (u) < lipf (u) + δL + δ < lipf (u) + ε.
It remains to proof that g is L + ε-Lipschitz. Obviously we have that
k∇g k ≤ L + ε.
Without loss of generality, we may assume that e(u) → 0 for u → ω for every
ω ∈ ∂U . So φ : U → R defined by
(
g(u), if u ∈ U,
φ(u) =
f (u), if u ∈ ∂U
is a continuous function. By Lemma 5, the function g is L + ε Lipschitz as well.
To prove Theorem 3 we need the following definable version of the Kirszbraun
Theorem.
Theorem 10 (Definable Kirszbraun Theorem). Let A ⊂ Rn be a definable set,
and let f : A → Rk be a definable L-Lipschitz map. Then there is a definable
L-Lipschitz map f : Rn → Rk such that f = f on A.
This Theorem has been proved by M. Aschenbrenner and the author in the far
more general context of definably complete structures over real closed fields in [1].
The use of f for the extension of f does not lead to confusion with the McShaneWhitney operator, as the Kirszbraun Theorem is a generalization to higher dimensional target spaces of the McShane-Whitney operator.
Proof of Theorem 3. Let ε0 > 0 be a rational number such that (1+ε0 )L < L+ε. By
[11, Thm. 1.9], the manifolds U and V admit definable C 1 tubular neighbourhoods
0
U 0 and V 0 . Let ρV : V 0 → V be the corresponding C 1 retraction
√ of V . If necessary,
0
0
shrink V , so that the derivative DρV of ρV is bounded by 1 + ε with respect to
the operator norm. Set
√
1 + ε0 − 1
√
.
δ :=
2 m
For i = 1, . . . , m, we denote by fi the McShane-Whitney L-Lipschitz extended
coordinate functions of f restricted to U 0 . Set f = (f1 , . . . , fm ). Then, for every
i = 1, . . . , m and for every definable continuous e0 : U 0 → (0, ∞), Theorem 2 implies
that there are definable (1 + δ) Lipschitz C 1 functions gi : U 0 → Rm which satisfy:
gi − fi < e0 on U 0 ,
√
lipgi ≤ lipfi +
1 + ε0 − 1
√
.
2 m
14
ANDREAS FISCHER
Set G = (g1 , . . . , gm ). Notice that for all u ∈ U 0 we have
√
lipG (u) < lipf (u) 1 + ε0 .
By choosing e0 sufficiently small we may assume that G(U ) ⊂ V 0 , and even that
|ρV ◦ G − f | < e on U.
Set g = ρ ◦ G restricted to U . Then g is (1 + ε0 )L Lipschitz be construction
Proof of Corollary 4. Let ε > 0 and L > 0. Let f : U → Rk be a bounded Lipschitz
map. Set
M := sup kf (u)k .
u∈U
Without loss of generality, we may assume that M > 0. Let δ > 0 be smaller than
min(ε/3, Lε/2M ). By Theorem 2 there is a definable C 1 -smooth L + δ Lipschitz
map g̃ : U → Rk such that |g̃ − f | < δ on U . Set
g :=
L
g̃.
L+δ
Then, the map g is L-Lipschitz, C 1 -smooth, and it satisfies
|g − f | ≤ |g − g̃| + |g̃ − f |
δ
+δ
L+δ
≤ ε/2 + ε/3
≤M
<ε
on U .
5. Remarks and Consequences
5.1. Let m ∈ N. Naturally the question arises whether more regularity of the
approximating function can be achieved. Shiota proved in [23] definable versions of
C 1 fine approximation of definable C 1 functions by definable C m functions so that C 1 regularity may be replaced by C m -regularity in Theorem 1, Theorem 2, Theorem
3 and Corollary 4. In the semialgebraic case, we may assume g to analytic; see
[22]. The author has proved in [15] that if the o-minimal structure admits C ∞
cell decomposition and is not polynomially bounded, then we may assume g to be
C ∞ -smooth.
5.2. As a consequence of Theorem 3 and Theorem 10 we obtain a C 1 -smooth definable Kirszbraun-Theorem:
Corollary 11 (Definable C 1 -smooth Kirszbraun Theorem). Let L > 0, and let
ε > 0. Let A ⊂ Rn be closed and definable, and let f : A → Rn be a definable LLipschitz function. Then there is a definable (L+ε)-Lipschitz function g : Rn → Rn
such that g = f on A and g is C 1 smooth outside of A.
APPROXIMATION OF O-MINIMAL MAPS SATISFYING A LIPSCHITZ CONDITION
15
5.3. Locally definable versions. There are several concepts of local o-minimality.
Here, we define it as follows. Let U ⊂ Rn be an open set. AS subset V of U is
called locally definable if for every u ∈ U there is an open ball B with center u such
that B ∩ V is definable. A function f : V → Rk is called locally definable if for
every u ∈ U there is an open ball B with center u such that f |B is definable. A
differentiable embedded manifold M ⊂ U is called locally definable if M is locally
definable as a subset of U . It then admits a locally finite atlas of definable charts.
This concept of local definability encompasses the setting of analytic-geometric
categories studied by van den Dries and Miller in [7]; in this case, the manifolds
are always analytic, and the underlying o-minimal structures are expansions of the
structure consisting of all global subanalytic sets. Our concept also generalizes
Shiota’s χ-setting; there, U is always the entire Euclidean space, see Shiota [23].
A further reason for our concept of local definability is motivated by the following
observation. Let U be an open subset of Rn , and let f : U → R be a continuous
function. For every continuous function e : U → (0, 1) there exists a locally finite
cover of U consisting of open balls Bi , i ∈ I for some index set I, such that
sup{|f (x) − f (y)| |x, y ∈ Bi } < inf{e(x)/m(i)|x ∈ Bi };
here m(i) denotes the number of balls having nonempty intersection with Bi . Let
xi denote the center of Bi . Choose for every i a semialgebraic continuous function
φi : U → R, which is strictly positive on Bi and which vanishes elsewhere. Define
g : U → R by
X φi (x)f (xi )
P
g(x) :=
j∈I φj
i∈I
Then |g − f | < e on U , and g is locally semialgebraic. In other words, the locally
semialgebraic (or definable) continuous functions on U are dense in the continuous functions with respect to the strong Whitney-topology. We do not know,
whether fine approximation of Lipschitz maps from Rn → Rm almost preserving
the Lipschitz-constant is possible. However, the locally definable setting appears
to be a good approximation of the non-definable setting.
From now on we always refer to our concept of local definability. Notice that,
independent from the underlying o-minimal structure, the locally definable continuous functions on U are dense in the continuous functions on U with respect to
the strong topology. So the rings of locally definable functions are a reasonably
good approximation of the general (i.e. not necessarily locally definable functions)
functions on U . However,
By a slight modification of our proofs - namely replacing finite by locally finite
- one obtains locally definable versions of Theorem 1, Theorem 2, Corollary 3,
Corollary 4 and Corollary 11.
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Gymnasium St. Ursula, Ursulastr. 8-10 46284 Dorsten, Germany
E-mail address: el.fischerandreas[at]live.de