#### Expressing the box cone radius in the relational calculus with real polynomial constraints

Discrete Comput Geom
Geometry Discrete & Computational
Floris Geerts 0
0 Department WNI, University of Limburg (LUC) , B-3590, Diepenbeek , Belgium
We show that there is a query expressible in first-order logic over the reals that returns, on any given semi-algebraic set A, for every point, a radius around which A is conical in every small enough box. We obtain this result by combining results from differential topology and real algebraic geometry, with recent algorithmic results by Rannou.
1. Introduction
The framework of constraint databases, introduced by Kanellakis et al. [
13
], provides a
nice theoretical model for spatial databases [
19
]. A spatial dataset is modeled using real
polynomial inequality constraints; such sets are also known as semi-algebraic sets [
1
],
[
4
]. The relational calculus (first-order logic) with real polynomial constraints then serves
as a basic query language, denoted here by FO.
The study of the expressive power of query languages for constraint databases is an
active domain of research [
18
]. One of the problems in particular that received attention
in recent years is that of determining the exact power of FO in expressing topological
properties of spatial databases [
3
], [
8
], [
11
], [
16
], [
21
]. One such well-known property,
which is central in this research, is that inside a small enough ball around each point, a
semi-algebraic set has the topology of a cone. A radius at which this behavior shows is
called a cone radius in the point for the set.
In this paper we prove a stronger property of semi-algebraic sets. We prove that inside
any box located around the center of and completely included in a small enough ball
around each point, a semi-algebraic set has the topology of a cone. A radius at which
this behavior shows is called a box cone radius in the point for the set. The existence of
such a radius was already proven for semi-algebraic sets in R2, but the methods of the
proof do not generalize to arbitrary dimensions [
16
].
Accordingly, a (box) cone radius query is a query that returns, for a semi-algebraic set
A in n-dimensional space Rn, a set of pairs ( p, r ) giving for every point p a (box) cone
radius r in p for A. In this paper we show that there exists an FO formula expressing a
(box) cone radius query. Again, the (box) cone radius query was shown to be expressible
in FO for semi-algebraic sets in R2, but the methods of the proof do not generalize to
arbitrary dimensions [
8
].
Expressibility of the (box) cone radius, apart from being a natural question in
itself, also has applications. Indeed, the expressive power of FO in expressing topological
properties is rather limited. For example, topological connectivity is not expressible in
FO [
2
]. Therefore, recursive extensions of FO have been studied in order to express
more queries [
12
], [
15
], [
17
], [
14
], [
7
], [
6
]. In particular, the question whether
topological connectivity is expressible in FO + TC, the extension of FO with a transitive
closure operator [
15
], was raised. This question was first answered affirmatively for
linear spatial databases in Rn [
15
]. Later, this result was extended to quadratic spatial
databases in R2 [
17
], and then to arbitrary closed spatial databases in R2 [
8
]. This last
result was obtained by expressing the cone radius query for closed spatial databases in
FO. In our companion paper [
9
] it is shown that for arbitrary spatial databases in Rn,
the expressibility of the box cone radius query implies that a piecewise linear spatial
database can be constructed in FO + TC, which has the same topological properties as
the original database. Hence, the question whether a spatial databases is connected, can
be reduced to the question whether a linear spatial database is connected. Since this last
question is expressible in FO + TC, topological connectivity of spatial databases in Rn
is also expressible in FO + TC. More generally, using the expressibility of the box cone
radius query, one can show that any computable topological query can be expressed in
FO + TCS, a variant of FO + TC in which one can control the termination behavior of
the transitive closure operator [
9
]. Again, this generalizes the result obtained for spatial
databases in R2 [
7
].
2. Preliminaries
2.1. Spatial Databases and Queries
A semi-algebraic set in Rn is a finite union of sets definable by conditions of the form
f1(x ) = f2(x ) = · · · = fk (x ) = 0,
g1(x ) > 0, g2(x ) > 0, . . . , g (x ) > 0,
with x = (x1, . . . , xn) ∈ Rn, and where f1(x ), . . . , fk (x ), g1(x ), . . . , g (x ) are
multivariate polynomials in the variables x1, . . . , xn with real coefficients. A database schema
S is a finite set of relation names, each with a given arity. A database over S assigns to
each S ∈ S a semi-algebraic set S D in Rn if n is the arity of S. A k-ary query over S is
a function mapping each database over S to a semi-algebraic set in Rk .
As query language we use first-order logic (FO) over the vocabulary (+, ·, 0, 1, <)
expanded with the relation names in S [
18
]. Let R be the model-theoretic structure
R, +, ·, 0, 1, < . A formula ϕ(x1, . . . , xk ) expresses the k-ary query defined by
ϕ(D) := {(a1, . . . , ak ) ∈ Rk | R, D |= ϕ(a1, . . . , ak )},
for any database D, where |= denotes the model-theoretic satisfaction relation. Note
that ϕ(D) is always semi-algebraic because all relations in D are; indeed, by Tarski’s
theorem [
24
], the relations that are first-order definable on the real ordered field are
precisely the semi-algebraic sets.
Example 1. The interior query is expressible in FO: Let S be a schema containing the
relation name S. Consider the FO formula
ϕint(x ) := (∃ε > 0)(∀x1) · · · (∀xn)( x − x
< ε → S(x1, . . . , xn)).
For any database D, ϕint(D) equals the interior of S D.
However, not every query is first-order expressible: the query which asks whether a
set is connected is not expressible in FO. This result and other results related to constraint
databases have recently been collected in a single volume [
18
].
2.2. Cones
Let A ⊆ Rn be a semi-algebraic set and let p ∈ Rn be a point not in A. We define the
cone with base A and top p as the union of all closed line segments between p and the
points in A. We denote this set by Cone( A, p) := {t b + (1 − t ) p | b ∈ A, 0 ≤ t ≤ 1}.
For a point p ∈ Rn and ε > 0, denote the closed ball centered at p with radius ε by
Bn( p, ε), and denote the sphere centered at p with radius ε, by Sn−1( p, ε).
We use the following notation: Let Rn be equipped with the standard Euclidean
topology. Let X ⊆ Y ⊆ Rn, the closure of X with respect to the induced topology on Y
is denoted by clY (X ), and intY (X ) indicates the interior of X with respect to the induced
topology on Y . When the ambient space Y is Rn, we omit the subscript Y . We denote
cl(X )\int(X ), the boundary of X , by bd(X ).
Let X ⊆ Rn and Y ⊆ Rm . A function h: X → Y is called a homeomorphism if it
is a bijection and both h and h−1 are continuous with respect to induced topology on X
and Y .
The following well-known theorem says that, locally around each point of A, a
semialgebraic set A has the topology of a cone.
Theorem [
1
], [
4
]. Let A ⊆ Rn be a semi-algebraic set and let p be a point of A. Then
there is a real number ε > 0 such that there exists a homeomorphism
h: A ∩ Bn( p, ε) → Cone( A ∩ Sn−1( p, ε), p).
Any real number ε > 0 as in the lemma is called a cone radius of A in p.
In this paper we prove the following theorem:
Theorem 1. Let A ⊆ Rn be a semi-algebraic set and let p be a point of A. Then there
is a real number ε > 0 such that for any box Box = [ p1 − b1, p1 + a1] × · · · ×
[ pn − bn, pn + an] ⊆ Bn( p, ε), there exists a homeomorphism
h: A ∩ Box → Cone( A ∩ bd(Box), p).
Any real number ε > 0 as in the theorem is called a box cone radius of A in p.
In Section 3 we show that every box cone radius is a cone radius. However, the
converse does not necessary hold as the following example shows:
Example 2. Consider A = {(x , y) ∈ R2 | (0 ≤ x ≤ 1 ∧ 0 ≤ y ≤ 1 ∧ x 2 + (y − 1)2 =
1) ∨ (x = 1 ∧ 1 ≤ y ≤ 3)} and let p = (0, 0). On the left of Fig. 1 the set A together
with three circles S1, S2, and S3 and two rectangles B1 and B2 are shown. On the right
of Fig. 1 the corresponding cones can be seen. It is clear that the radius of S1 is not a
cone radius of A in p (the cone is the empty set) and that the radius of both S2 and S3 is
a cone radius of A in p. However, looking at the cone corresponding to B1, the radius
of S2 is not a box cone radius of A in p (the cone is a two-dimensional set). The radius
of S3 is a box radius of A in p as can be seen from the cone corresponding to B2.
Let S be a schema containing a relation name S of arity n. A (box) cone radius query
Qradius is a query which maps any database D over S to a set of pairs ( p, r ) ∈ Rn × R
such that for every point p ∈ S D there exists at least one pair ( p, r ) ∈ Qradius(D), and
for every ( p, r ) ∈ Qradius(D), r is a (box) cone radius in p for S D.
Our second result is the following:
Theorem 2. There exists an FO-expressible box cone radius query.
2.3. C 1-Whitney Decomposition
In this section we construct a C 1-Whitney decomposition of a semi-algebraic set A and
show that this construction is expressible in FO. The construction consists of several steps.
Firstly, A is decomposed in parts which are C 1-smooth, resulting in the C 1-decomposition
Expressing the Box Cone Radius in the Relational Calculus with Real Polynomial Constraints
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of A. Secondly, the decomposition is refined such that all points in a single part have the
same local topological type. This gives the C 1-Whitney decomposition of A. Finally,
this decomposition is made compatible with a finite number of specified sets.
Before giving the formal definition of a C 1-decomposition, we give an example.
Example 3. Consider the semi-algebraic set A = {(x , y) ∈ R2 | (y > |x |) ∨ (y =
−x ∧ y ≥ 0)} which is depicted in Fig. 2. We first look at those points where A has
no tangent space. It is clear that A has no tangent space in the origin. However, if we
decompose A into {(x , y) ∈ R2 | (y > |x |) ∨ (y = −x ∧ y > 0)} and {(0, 0)}, then both
parts have a tangent space in any of its points. We then decompose these parts according
to their dimension: let A2 be the two-dimensional part {(x , y) ∈ R2 | y > |x |}, let A1
be the one-dimensional part {(x , y) ∈ R2 | (y = −x ∧ y > 0)}, and finally let A0 be
the zero-dimensional part {(0, 0)}. The sets A0, A1, A2 will form the C 1-decomposition
of A.
The following definitions are taken from [
20
]. Let A be a semi-algebraic set in Rn.
The secants limit set of A in a point p ∈ A is defined as the set
limsecp A :=
cl({λ(u − v) ∈ Rn | λ ∈ R and u, v ∈ A ∩ Bn( p, η)}).
η>0
If limsecp A is a vector space (this is true when for all s, t ∈ limsecp A, the sum
s + t is also an element of limsecp A), then we define the tangent space of A in p as
Tp A := p + limsecp A. If limsecp A is not a vector space, the tangent space of A in p
is undefined. The set A is C 1-smooth in p if and only if Tp A exists and there exist a
neighborhood U of p such that the orthogonal projection of A ∩ U on Tp A is bijective.
A set is C 1-smooth if it is C 1-smooth in all its points. We define the set
Smoothk ( A) = {x ∈ Rn | A is C 1-smooth in x and of dimension k}.
We can now decompose [
26
], [
22
] A into at most n + 1 nonempty C 1-smooth parts
A0, . . . , An as follows: Define An = Smoothn( A). Suppose that An, . . . , Ak+1 is already
constructed. Then define
Ak = Smoothk
A\
n
i=k+1
Ai .
(1)
The sets A0, . . . , An are called the C 1-decomposition of A.
F. Geerts
Let S be a database schema containing a relation name S of arity n. For each k ≥ 0,
define the query Qk-smooth as Qk-smooth(D) := Smoothk (S D) for any database D over S.
By the constructions given by Rannou [
20
], the following is readily verified:
Proposition 1. Let S be a database schema containing a relation name S of arity n.
For each 0 ≤ k ≤ n the query Qk-smooth is expressible in FO.
The following example motivates the construction of the C 1-Whitney decomposition.
Example 4. Let A = {(x , y, z) ∈ R3 | x 2 − zy2 = 0}. This set is known as the Whitney
umbrella and is depicted in Fig. 3. The C 1-decomposition of A consists of two nonempty
sets A2 = Smooth2( A) = {(x , y, z) ∈ R3 | x 2 − zy2 = 0 ∧ ¬(x = 0 ∧ y = 0)} and
A1 = Smooth1( A) = {(x , y, z) ∈ R3 | x = 0 ∧ y = 0}. However, the local topological
type changes when one looks at points in A1 with z < 0, z = 0, and z > 0. On the
contrary, any two points in A2 have the same local topological type. For this reason, one
aAg12re=es {t(ox s,pyl,itz)up∈thRe3se|txA=1in0to∧ Ay11==0{(∧x ,z y>,z)0}∈, aRnd3 |A0x == {0(x∧, yy, z=) ∈0 ∧R3z |<x 0=},
0 ∧ y = 0 ∧ z = 0}. Then the sets A0, A1 1
1 ∪ A2, and A2 form a decomposition of A
which is called the C 1-Whitney decomposition of A.
To avoid such situations as that in Example 4, Whitney [
27
] introduced the following
condition for C 1-smooth semi-algebraic sets X, Y ⊆ Rn and a point x ∈ X .
One says that the triple (X, x , Y ) has the Whitney property when the following holds:
if (xi ), (yi ) are sequences in X , Y , respectively, both converging to x , if the sequence
of tangent spaces (Tyi Y ) converges to a subspace T ⊆ Rn, and if the sequence (−xi→yi ) of
lines containing xi − yi converges to a line L ⊆ Rn, then L ⊆ T . One says that (X, Y )
has the Whitney property if (X, x , Y ) has the Whitney property for any point x ∈ X . We
define the set Whitney(X, Y ) = {x ∈ Rn | X, Y are C 1-smooth, x ∈ X and (X, x , Y )
has the Whitney property}.
Let S be a database schema containing two relation names S1 and S2 of arity n. Define
the n-ary query, defined as QWhitney(D) := Whitney(S1D, S2D) for any database D over S.
Expressing the Box Cone Radius in the Relational Calculus with Real Polynomial Constraints
613
Again, by constructions given by Rannou [
20
] the following is readily verified:
Proposition 2. Let S be a database schema containing two relation names S1 and S2
of arity n. The n-ary query QWhitney is expressible in FO.
We can decompose A into n + 1 C 1-smooth sets A0, . . . , An such that ( Ai , Aj ) has
the Whitney property for every i < j . Indeed, let An := Smoothn( A). Now suppose
An, . . . , Ak+1 have already been constructed. Part Ak is then constructed as follows:
Rk := Smoothk
A\
Ak :=
n
i=k+1
n
i=k+1
Ai ,
intRk (Whitney(Rk , Ri )).
(2)
(3)
In (3) the interior is taken relative to the set Rk . As a result, the set Ak is still C 1-smooth
because it is an open subset of the C 1-smooth set Rk . The sets A0, . . . , An are called the
C 1-Whitney decomposition of A.
A C 1-Whitney decomposition A0, . . . , An of A is called compatible with a finite set
of semi-algebraic sets {B1, . . . , Bk } if for any of the Bi ’s, each connected component of
the Ai ’s is either included in or disjoint with Bi .
For reasons that will become clear in Section 3, we now construct a C 1-Whitney
decomposition of cl( A) which is compatible with A and {x ∈ Rn | x1a1 p1, . . . , xnan pn}
for each tuple (a1, . . . , an) ∈ {=, =}n.
Example 5. Consider again the semi-algebraic set A = {(x , y) ∈ R2 | (y > |x |) ∨
(y = −x ∧ y ≥ 0)}. Let A2 = {(x , y) ∈ R2 | y > |x | > 0}, A1 = {(x , y) ∈ R2 | y =
−x ∧ y > 0} ∪ {(x , y) ∈ R2 | y = x ∧ y > 0} ∪ {(x , y) ∈ R2 | x = 0 ∧ y > 0}, and
A0 = {(0, 0)}. Then A0, A1, A2 is a C 1-Whitney decomposition of cl( A) compatible
with A, {(0, 0)}, {(x , y) ∈ R2 | x = 0}, and {(x , y) ∈ R2 | y = 0}. This example is
illustrated in Fig. 4.
We now decompose cl( A) into n + 1 C 1-smooth parts A0, . . . , An such that ( Ai , Aj )
has the Whitney property for every i < j , and such that this decomposition is compatible
with A and {x ∈ Rn | x1a1 p1, . . . , xnan pn} for each tuple (a1, . . . , an) ∈ {=, =}n.
The following construction is an adaptation of the construction given by
Shiota [22, Lemma I.2.2].
We define An = Smoothn( A). Now suppose that the parts An, . . . , Ak+1 have already
been constructed. Then part Ak is constructed as follows. Define B0 = A, B1 = cl( A)\ A
and for i = 0, 1 define
Biσ1···σn := {x ∈ Bi | x1σ1 p1, . . . , xnσn pn},
with σ1, . . . , σn ∈ {<, =, >}. For each tuple σ = (σ1, . . . , σn) ∈ {<, =, >}n and
i = 0, 1 construct
Riσ,k := Smoothk
Biσ\
n
j=k+1
Wiσ,k :=
intRiσ,k (Whitney(Riσ,k , Aj )),
n
j=k+1
Aj ,
Aiσ,k := Wiσ,k \cl W1σ−i,k ∪
σ ∈{<,=,>}n
σ =σ
(W0σ,k ∪ W1σ,k ) .
(4)
(5)
(6)
Then we define Akσ = A0σ,k ∪ A1σ,k and Ak := σ∈{<,=,>}n Akσ. Set Ak indeed has
the desired properties: by (4) it is C 1-smooth and of dimension k, (5) guarantees
that for all points in Ak , and for any j > k, ( Ak , Aj ) has the Whitney property, and
(6) ensures that the connected components are either included in or disjoint with A and
{x ∈ Rn | x1a1 p1, . . . , xnan pn} for each tuple (a1, . . . , an) ∈ {=, =}n.
It is well known [
26
], [
22
] that the dimension of cl( A)\ nj=k Aj is strictly smaller
than the dimension of cl( A)\ nj=k+1 Aj for i = 0, 1. Hence, the decomposition will
consists of exactly n + 1 sets Ak , some of which may be empty.
Let S be a database schema containing a relation name S of arity n. We define the
nary query Qk-part(D) = (S D)k , with (S D)k the kth part of the decomposition constructed
above for A = S D.
A direct consequence of Propositions 1 and 2 is the following:
Proposition 3. Let S be a database schema containing a relation name S of arity n.
For each 0 ≤ k ≤ n, the n-ary query Qk-part is expressible in FO.
3. Expressing the Box Cone Radius in FO
Before proving Theorem 1, we look again at Example 2. In this example it is clear what
the local topology of A in p is. Indeed, A looks like a straight line locally around p.
Hence, a radius ε will be a box cone radius if the boundary of any box around p in
Bn( p, ε) intersects A in a single point. For this reason the radius of S2 is not a box
cone radius (there is even a box whose boundary has a one-dimensional intersection!).
Similarly, if ε is large and you take a very large box, the intersection of the boundary of
this box with A will be empty. Also in this case, ε will not be a box cone radius.
Expressing the Box Cone Radius in the Relational Calculus with Real Polynomial Constraints
615
Since the boundary of a box in R2 consists of horizontal and vertical line segments, it is
sufficient to look at the intersection of those segments with A. The possible intersections
are shown in Fig. 5. Only cases (b) and (e) deliver a single intersection point. In the proof
of Theorem 1 we will identify the points in which the intersection behavior changes for
the vertical or horizontal line segments. In Example 2 the vertical intersections change
from the empty set to one point, when crossing the point (1, 3). Similarly, the horizontal
intersections change from the empty set to a single point when crossing the horizontal line
at x = 1. The points in which these changes occur will be the so-called critical points.
These points correspond to zero-dimensional parts of the C 1-Whitney decomposition
(like the point (1, 3)), or those points where there is a horizontal or vertical tangent
space (like the points in {(x , y) ∈ R2
| x = 1 ∧ (1 ≤ y < 3)}). A radius ε will then be
a box cone radius when B2( p, ε) does not contain any critical points. The critical point
closest to p is the point (1, 1), hence any radius smaller than 1 will be a box cone radius.
We will formalize the above intuitive example and prove the first result of this paper.
Theorem 1.
[ pn − bn, pn + an] ⊆ Bn( p, ε), there exists a homeomorphism
is a real number ε > 0 such that for any box Box = [ p1 − b1, p1 + a1] × · · · ×
Let A ⊆ Rn be a semi-algebraic set and let p be a point of A. Then there
h: A ∩ Box → Cone( A ∩ bd(Box), p).
with fj−1(0) for any j = 1, . . . , n.
( pn − xn)2 and fn+1(x ) = ( p1 − x1)2 + · · · + ( pn − xn)2.
Proof.
Let A ⊆
Rn, p ∈ A, and let A0, . . . , An be the C 1-Whitney
decomposi(a1, . . . , an) ∈ {=, =}n. Let x ∈
tion of cl( A) compatible with A and {x ∈
Rn and define f1(x ) = ( p1 − x1)2, . . . , fn(x ) =
| x1a1 p1, . . . , xnan pn} for each tuple
For each k = 0, . . . , n, any connected component of Ak is the disjoint union of
connected components of Akσ with σ ∈ {<, =, >}n. Since the C 1-Whitney
decomposition A0, . . . , An is compatible with {
x ∈
| x1a1 p1, . . . , xnan pn} for each tuple
(a1, . . . , an) ∈ {=, =}n, any connected component of Akσ is either included in or disjoint
Rn
Rn
those indices j such that Aσ
k ∩ fj−1(0) = ∅
.
For each k = 1, . . . , n and any σ ∈ {<, =, >}n, we define Jkσ ⊆ {1, . . . , n + 1} as
The theorem directly follows from the following two claims:
Claim 1. If ε is a positive real number such that
( A0\{ p}) ∩ int(Bn( p, ε)) = ∅
(7)
and, for each k = 1, . . . , n and any σ ∈ {<, =, >}n and j ∈ Jkσ, the restriction
fj | ( Akσ ∩ int(Bn( p, ε))) −→ R has no critical points,
then ε is a box cone radius in p for A.
Here, the critical points of fj | X for some subset X ⊆ Rn, are the points x ∈ X where
the differential mapping dx ( fj | X ): Tx X → R, defined by dx ( fj | X )(v1, . . . , vn) =
((∂( fj | X )/∂ x1)(x ), . . . , (∂( fj | X )/∂ xn)(x )) · (v1, . . . , vn), is not surjective.
Claim 2. There exists a positive real number ε such that the conditions of the above
claim hold.
Proof of Claim 1. We equip each part Ak with the standard Riemannian metric ,
induced from Rk . Then for any C 1-function f on Ak , we can define the gradient vector
field grad( f | Ak ). The value of grad( f | Ak ) in a point x is the unique vector in Tx Ak
with the property
grad( f | Ak )(x ), v = dx ( f | Ak )(v).
Next, for each σ ∈ {<, =, >}n, we define the following continuous vector field on Aσ:
k
ξkσ = −grad( fn+1 | Akσ)
j∈Jkσ\{n+1}
|grad( fj | Akσ)|.
Clearly, ξkσ(x ) = 0 if and only if x ∈ Akσ is a critical point of at least one of the fj | Akσ’s
with j ∈ Jkσ. In Fig. 6 we have depicted the vector field ξ in case where A = R2 and
p = 0.
By conditions (7) and (8), ε > 0 is such that within int(Bn( p, ε)) no critical points
occur. Hence, ξkσ(x )( fj ) < 0 for x ∈ Akσ ∩ int(Bn( p, ε)) and each j ∈ Jkσ. (Here, we
take the alternative view of tangent vectors where the vector ξkσ(x ) is seen as a function
which maps real-valued functions on Akσ to a real number [5, Chapter 8].)
Since parts A1, . . . , An are C 1-manifolds, we can clearly obtain a continuous flow
on each part Aiσ by integrating the continuous vector field ξkσ. In general, however, we
cannot expect to obtain a continuous flow on the set cl( A) by just putting the flows
(8)
Expressing the Box Cone Radius in the Relational Calculus with Real Polynomial Constraints
617
together. Therefore the vector fields ξkσ will be transformed into vector fields ηkσ which
are controlled along the boundary of each part. Integrating these new vector fields and
putting the flows together then results in a continuous flow on cl( A) [
10
].
Technically, the controlled vectors field ηkσ are constructed inductively on the
dimension of the parts by means of a C 1-controlled tube system for the C 1-Whitney
decomposition A1, . . . , An [
23
], [22, Lemmas I.1.3 and I.1.5]. Moreover, these vector fields can be
chosen such that they also satisfy ηkσ(x )( fj ) < 0 for x ∈ Akσ ∩ int(Bn( p, ε)) and j ∈ Jkσ.
Since cl( A) ∩ int(Bn( p, ε)) is locally closed, the vector fields ηkσ admit a continuous
flow θiσ which nicely fit together into a continuous flow on cl( A) ∩ int (Bn( p, ε)) [22,
Lemma I.1.6].
Let Box = [ p1 − b1, p1 + a1] × · · · × [ pn − bn, pn + an] ⊆ int(Bn( p, ε)). We
will show that the flow induces the required homeomorphism between A ∩ Box and
Cone( A ∩ bd(Box), p), proving that ε is a box cone radius in p for A.
The flow is the union of flows θkσ for k = 1, . . . , n and σ ∈ {<, =, >}n. Each
individual flow θkσ can be chosen to be the continuous map
R × ( Akσ ∩ int(Bn( p, ε))) → Aσ
k ∩ int(Bn( p, ε)),
which is the unique solution of
d
dt θkσ(t, x ) = ηkσ(x )
k ∩ int(Bn( p, ε)) and initial condition θkσ(0, x ) = x for x ∈ Akσ ∩ bd(Box).
for x ∈ Aσ
Since ηkσ(x )( fj ) < 0 for j ∈ Jkσ, we have that
| pj − (θkσ(t1, x ))j | > | pj − (θkσ(t2, x ))j |
for any t1 < t2,
(9)
for all j ∈ Jkσ. This means that the coordinates of the integral curves γx : t → θkσ(t, x )
are either monotone decreasing for increasing values of t , or equal to pj for some j .
For each t ∈ (0, 1], define the box Boxt = [ p1 − t b1, p1 + t a1] × · · · × [ pn − t bn, pn +
t an] and the mapping (illustrated in Fig. 7)
(hkσ)t : Aσ
k ∩ bd(Box) → ( Akσ ∩ bd(Boxt ))\{ p},
x → γx ∩ bd(Boxt ).
We now define ht : cl( A) ∩ bd(Box) → cl( A) ∩ bd(Boxt ) by ht (x ) = (hkσ)t (x ), where
k and σ are the unique indices such that x ∈ Akσ, and show that it is a homeomorphism
for each t ∈ (0, 1]. It suffices to prove that each (hkσ)t is a function, since bijectivity and
continuity follow directly from the properties of flows of vector fields [
10
].
Firstly, for each x ∈ Aσ k ∩ bd(Boxt ) such that
k ∩ bd(Box) there exists a point y ∈ Aσ
(hkσ)t (x ) = y. Indeed, it follows from property (9) that limt →+∞ θkσ(t , x ) = p, and,
hence, the curve γx must have an intersection with bd(Boxt ).
Secondly, suppose that for an x ∈ Aσ
k ∩ bd(Box), (hkσ)t (x ) = {y1, y2}. By definition
of (hkσ)t , this happens when θkσ(t1, x ) = y1 and θkσ(t2, x ) = y2, or, in other words,
when the integral curve γx intersects bd(Boxt ) twice. Clearly, property (9) implies that
y1 = y2. However, we show that this also is impossible. We give the argument for n = 2,
the n-dimensional case being analogous.
In this case, Boxt = [ p1 − t b1, p1 + t a1] × [ p2 − t b2, p2 + t a2]. By property (9), there
exists a unique σ ∈ {<, =, >}2 such that θkσ(t, x ) ⊆ Boxtσ.1 Suppose that θkσ(t, x ) ⊆
Bt(>,>), the other cases being analogous. Hence, either
1. y1 = ( p1 + t a1, p2 + t a2); or
2. y1 = ( p1 + t a1, u) with p2 < u ≤ p2 + t a2; or
3. y1 = (v, p2 + t a2) with p1 < v ≤ p1 + t a1.
Suppose that we are in the second case. Then y2 = (u , v ) with p1 < u < p1 + t a1
and p2 < v < u. Since y2 ∈ Boxt , v must be equal to p2 + t a2, but this is impossible
since we have that p2 < v ≤ u ≤ p2 + t a2. Hence, (hkσ)t is a function.
Next, we define the homeomorphism
h1 := (0, 1] × (cl( A) ∩ bd(Box)) → cl( A) ∩ Box\{ p},
by h1(t, x ) := ht (x ).
Since A0, . . . , An is compatible with A and the flows of which h1 is constructed
preserve the connected components of the Ak ’s, the restriction h1−1 | (( A ∩ Box)\{ p})
will also be a homeomorphism between ( A ∩Box)\{ p} and (0, 1]×( A ∩bd(Box)). Since
the cylinder (0, 1] × ( A ∩ bd(Box)) is homeomorphic to Cone( A ∩ bd(Box), p)\{ p},
e.g., by the homeomorphism
h2(t, x ) := (1 − t ) p + t x ,
we obtain a homeomorphism
h3 := h2 ◦ h1−1: ( A ∩ Box)\{ p} → Cone( A ∩ bd(Box), p)\{ p}.
The homeomorphism h3 can be trivially extended to the point p, resulting in the desired
homeomorphism h.
Proof of Claim 2. We now observe that for each p there exists a positive real number
ε > 0 such that both (7) and (8) are satisfied.
Indeed, it is clear that the distance between p and A0\{ p} is strictly positive. Let ε0
be this distance.
1 Let X ⊆ Rn and σ ∈ {<, =, >}n. Then Xσ = {x ∈ X | x σ1 0, . . . , x σn 0}.
Next, a critical value of fj | Akσ is the image by fj | Akσ of a critical point. The
set of critical points of fj | Akσ is semi-algebraic and admits a C 1-cell decomposition
C = {C1, . . . , Cm } such that fj | C is C 1 [
25
]. Sard’s theorem for C 1-mapping [
28
]
implies that each fj | C attains only a finite number of values. Hence the image by
fj | Akσ of the set of critical points is finite. This is true for every k = 1, . . . , n and every
σ ∈ {<, =, >}n and j ∈ Jkσ.
Denote by εσjk the minimal critical value of fj | Akσ. (If there are no critical values,
k ∩ fj−1(0). However, this
set εσjk = 1.) Note that εσjk can only be zero for points of Aσ
implies that j ∈ Jkσ, which is impossible. Hence, any 0 < ε < min{ε0, εσjk /n | k =
1, . . . , n, σ ∈ {<, =, >}n, j ∈ Jkσ} is a good one.
This concludes the proof of Theorem 1.
Theorem 2. There exists an FO-expressible box cone radius query.
Proof. Let S be a schema containing a relation name S of arity n, and let D be a
database over S. By Claim 1, we can define the following cone radius query:
Qradius(D) := {( p, r ) ∈ Rn × R | p ∈ S D and r ∈ (0, ε)},
where ε is such that conditions (7) and (8) are satisfied for the semi-algebraic set A = S D.
Let us express this query in FO.
We define the critical point query as
Qcrit(D) := {( p, x ) ∈ Rn × Rn | x ∈ Q0-Whitney(D) or p ∈ S D
and x ∈ Qk-Whitney(D) for a some k, fj (x ) = 0 and
x is a critical point of fj | Qk-Whitney(D)
for a certain j = 1, . . . , n + 1}.
Claim. Let Z be a C 1 semi-algebraic set in Rn of dimension k. Then x ∈ Rn is a
critical point of fj | Z if and only if the tangent space of Z is parallel to {(x1, . . . , xn) ∈
Rn | xj = 0}. Similarly, x ∈ Rn is a critical point of fn+1 | Z if and only if the tangent
space of Z in x is orthogonal to p − x .
Proof of Claim. We prove the claim for f = fn+1 = ( p1 − x1)2 + · · · + ( pn − xn)2, the
other case being subcases. We compute the differential dx ( f | Z ) as follows: Locally
around x , we may assume that the projection on the first k coordinates : Z → U ⊆ Rk
is a homeomorphism. By definition of the differential, dx ( f | Z ) = (d(x1,...,xk)g)
(d(x1,...,xk) −1)−1, where g = ( f | Z ) ◦ −1. By the C 1 Inverse Function Theorem,
we may assume that −1: U → Z : (x1, . . . , xk ) → (x , . . . , xk , ϕk+1, . . . , ϕn), where
ϕxii)(2x1+, . . .nj,=xkk+)1(aprej −C 1ϕ-jm(xa1p,p.in..g,s,xka)n)d2.hAennceelegm: eUnta→ry caRlc:u(lxa1ti,o.n. .sh,oxwk)s =that t
hik=e1d(ipffie−rn
j=k+1
∂ϕj (x1, . . . , xk )
( pj − xj ) ∂ xi
ential of f | Z in x is the vector
dx ( f | Z ) = 2 ( pi − xi ) +
(v1, . . . , vn) =
Hence, the product
Since d(x1,...,xk) −1 is an isomorphism between the tangent space T(x1,...,xk)U of U in the
projection (x ), and the tangent space Tx Z of Z in x , any tangent vector (v1, . . . , vn) ∈
Tx Z is of the form (d(x1,...,xk) −1)(v1, . . . , vk ). More specifically, any tangent vector
v ∈ Tx Z can be written as
v1, . . . , vk , k ∂ϕk+1 (x1, . . . , xk )vi , . . . , k ∂ϕn (x1, . . . , xk )vi .
i=1 ∂ xi i=1 ∂ xi
∧
n
n
j=1 k=1
k
i=1
dx ( f | Z )v = 2
( pi − xi )vi + 2
( pj − xj )
n
j=k+1
k ∂ϕj (x1, . . . , xk )vi
i=1 ∂ xi
is equal to 2 in=1(xi − pi )vi . This implies that the differential mapping dx ( f | Z ) is
not surjective if and only if 2 in=1( pi − xi )vi = 0 for all tangent vectors v ∈ Tx Z . It is
clear that this implies the claim.
The proof of the theorem now continues as follows. The tangent space query
Qtangent(D) := {(x , v) ∈ Rn × Rn | S D is C 1, x ∈ S D and v ∈ Tx S D}
is expressible in FO [20, Lemma 2]. Because the orthogonality and parallelism of two
vectors can be easily expressed in FO, the formula
ϕcrit( p, x ) = (ϕ0-part(S)(x ) ∧ (x = p)) ∨ S( p)
(∀vϕtangent(ϕj-part(S))(x , v) ∧ (xk = pk ) → ( pk − xk )vk = 0)
∨ ∀vϕtangent(ϕj-part(S))(x , v) ∧ (x = p)
→ ( p1 − x1)v1 + · · · + ( pn − xn)vn = 0
expresses Qcrit correctly by the above claim. Here, ϕj-part denotes an FO formula
expressing Qj-part for j = 0, . . . , n, and ϕtangent is an FO formula expressing Qtangent.
Let ϕval( p, r ) be the FO formula which expresses the query which returns the critical
values of critical points given by Qcrit.
By the above there exists a minimal critical value and any value smaller than this
minimal value is a cone radius. We therefore conclude that the query expressed in FO as
ϕradius( p, r ) := (∀r )(ϕval( p, r ) → r < r )
is a cone radius query, as desired.
Theorem 3.
There exists an FO-expressible cone radius query.
Proof. The proof is the same as for Theorems 1 and 2, except that the constructed
C 1-Whitney decomposition of cl( A) only needs to be compatible with A and only the
critical points with respect to fn+1 must be considered.
Corollary 1.
Every box cone radius is a cone radius.
Proof. This follows from the fact that the set of critical points identified in the proof
of Theorem 1 is a superset of those identified in the proof of Theorem 3.
Acknowledgments References
The author thanks Jan Van den Bussche for many interesting discussions and critical
remarks, which led to improvements in the presentation of the paper. The author also
thanks Masahiro Shiota for his feedback on the construction of the vector fields in the
proof of Theorem 1.
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