Generalizing the Poincaré – Miranda theorem:
the avoiding cones condition
Alessandro Fondaa,˚, Paolo Gidonib
a
Dipartimento di Matematica e Geoscienze, Università di Trieste,
P.le Europa 1, I-34127 Trieste, Italy
b
SISSA - International School for Advanced Studies,
Via Bonomea 265, I-34136 Trieste, Italy
Abstract
After proposing a variant of the Poincaré–Bohl theorem, we extend the
Poincaré–Miranda theorem in several directions, by introducing an avoiding
cones condition. We are thus able to deal with functions defined on various
types of convex domains, and situations where the topological degree may
be different from ˘1. An illustrative application is provided for the study of
functionals having degenerate multi-saddle points.
Keywords: Poincaré–Miranda, fixed point, topological degree
2010 MSC: 47H10, 54H25, 37C25
1 Introduction
Let us start recalling the Poincaré–Miranda theorem. We consider a
rectangle R in RN ,
R “ ra1 , b1 s ˆ ¨ ¨ ¨ ˆ raN , bN s ,
I
Acknowledgement. The authors have been supported by the Gruppo Nazionale per
l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto
Nazionale di Alta Matematica (INdAM).
˚
Corresponding author
Email addresses: [email protected] (Alessandro Fonda), [email protected] (Paolo
Gidoni)
Version: February 10, 2015
and a continuous function f : R Ñ RN . For every x P R, we write
f pxq “ pf1 pxq, . . . , fN pxqq ,
thus defining the components fk : R Ñ R, with k “ 1, . . . , N . The theorem
states that, if each component fk changes sign on the corresponding opposite
faces
Fk´ “ tx P R : xk “ ak u , Fk` “ tx P R : xk “ bk u ,
then there is a point in R where all the components of f are equal to zero.
Theorem 1 (Poincaré–Miranda). Assume that, for k “ 1, . . . , N , either
"
ď 0 , for every x P Fk´ ,
fk pxq
(1a)
ě 0 , for every x P Fk` ,
or
"
fk pxq
ě 0 , for every x P Fk´ ,
ď 0 , for every x P Fk` .
(1b)
Then, there is an x̄ P R such that f px̄q “ 0.
As recalled in the nice survey paper [18] (see also [16]), this theorem was
stated in 1883 by Jules Henri Poincaré [23], with just a hint of proof, and
then forgotten for a long time. In 1940, Silvio Cinquini [4] rediscovered its
statement, but his proof was not complete, and it was one year later that
Carlo Miranda [20] proved the equivalence of Theorem 1 with Brouwer’s fixed
point theorem.
Poincaré was mainly motivated by the study of periodic solutions for
differential equations arising from the three body problem, which can be
obtained as fixed points of what is now called the Poincaré map. After his
pioneering work, many other researchers found applications of Theorem 1,
and different generalizations have been proposed, see e.g. [14, 19, 21, 22, 26,
27, 29, 30, 31, 32].
The aim of this paper is to generalize Theorem 1 by replacing the sign
condition on the components of the vector field with an avoiding cones condition. We will also consider different shapes for the domain of the function f ,
instead of just rectangles, leading to situations where the topological degree
can be different from ˘1. The starting point to obtain these results will be
a variant of the Poincaré–Bohl theorem.
2
When the domain D of the function f is a convex body and contains the
origin in its interior, our modification of the Poincaré–Bohl theorem consists
in replacing the usual assumption, requiring the vector field to avoid the
rays arising from the origin, by asking instead that it avoids the normal
cones. Clearly enough, when the boundary of the set is smooth, this means
avoiding the rays determined by the normal vector field: in this case, such a
condition has been tackled in [11]. The modified Poincaré–Bohl theorem will
be introduced in Section 2, while a first generalization of Theorem 1 will be
given in Section 3.
We then propose in Section 4 a different viewpoint, interpreting the domain D as a truncated convex body. This leads us to the problem of reconstructing a larger convex body E where to extend our function f . The set
E is obtained by glueing D with some convex sets C1 , . . . , CM , where M
denotes the number of truncations. In Section 5 we argue about the optimal
choice of the reconstruction E, which minimizes the amplitude of the cones
to be avoided by the vector field f .
In Section 6 we state our main result for truncated convex bodies. Here,
the avoiding cones condition is stated in its most general form, assuming that
the field f avoids the inner normal cones in the regions of the boundary of D
where the truncation occurs, and the outer normal cones on the remaining
part of the boundary. In this situation, we show that the topological degree
of f on D is equal to ˘p1 ´ M q. This will proved in Section 8, by suitably
extending the function f to the larger set E “ D Y C1 Y ¨ ¨ ¨ Y CM , and using
the additivity of the degree. Then we show how the convexity assumption on
the set D can be weakened, by only requiring the set D to be diffeomorphic
to a convex body.
The structure of the Poincaré–Miranda theorem is typical of the gradient
of a potential V in a neighbourhood of a non-degenerate saddle point, where
the domain is split in two subspaces, in such a way that the vector field
is expansive on one subspace and contractive on the other one [18, 21, 22].
More generally, also degenerate saddle points can be characterized by their
associated expansive and contractive directions.
In Section 7, we provide an illustrative example showing how our results
apply to such situations, considering the function f “ ∇V in a neighbourhood of a degenerate multi-saddle point. Indeed, we can deal with some more
general situations, and show that the degree of ∇V is equal to ˘p1 ´ M q,
3
where M is the number of connected components of the set of boundary
points where ∇V points outwards. This type of results has already been
treated by many authors (see, e.g., [6, 25, 28] and the references therein).
2 A variant of the Poincaré–Bohl theorem
We would like to discuss about the well known Poincaré–Bohl theorem,
which we recall, together with its proof, for the reader’s convenience.
Theorem 2 (Poincaré–Bohl). Assume that Ω is an open bounded subset of
RN , with 0 P Ω, and that f : Ω Ñ RN is a continuous function such that
f pxq R tαx : α ą 0u ,
for every x P BΩ .
(2)
Then, there is an x̄ P Ω such that f px̄q “ 0.
Proof. We may assume that f pxq ‰ 0 for every x P BΩ, since otherwise there
is nothing more to prove. Consider the homotopy F : Ωˆr0, 1s Ñ RN defined
by
F px, λq “ p1 ´ λqf pxq ´ λx .
We show that 0 R F pBΩ ˆ r0, 1sq. By contradiction, assume that there are
an x P BΩ and a λ P r0, 1s such that F px, λq “ 0. Then λ ‰ 0, since by
the above assumption f pxq ‰ 0, and λ ‰ 1, since 0 P Ω. So, λ P s0, 1r and,
setting α “ λ{p1 ´ λq, we see that α ą 0, and f pxq “ αx, a contradiction.
Therefore, we can compute the Brouwer topological degree:
dB pf, Ωq “ dB pF p¨, 1q, Ωq “ dB pF p¨, 0q, Ωq “ dB p´I, Ωq “ p´1qN .
(Here and in the following, I denotes the identity function.) The conclusion
readily follows.
We now state a variant of Theorem 2, in the case when the set D “ Ω
is a convex body, i.e., a compact convex set in RN with non-empty interior.
Note that every convex body coincides with the closure of its interior.
4
x
x
D
D
0
(a) Poincaré–Bohl condition (2).
(b) Avoiding outer cones condition (4).
Figure 1: Comparison between condition (2) of the Poincaré–Bohl
theorem and the avoiding outer cones condition (4). The red halflines
and cones indicate the regions avoided by f pxq for some x P BD.
Given a point x̄ P D, we define the normal cone to D in x̄ as
(
ND px̄q “ v P RN : xv, x ´ x̄y ď 0 , for every x P D ,
(3)
where, as usual, x¨ , ¨y denotes the euclidean scalar product in RN , with associated norm k¨k. Trivially, ND pxq “ t0u for every x P int D. On the other
hand, it can be shown that, if x P BD, then its normal cone contains at least
a halfline. For x P BD, we write ND0 pxq to denote ND pxq deprived of the
origin, i.e., ND0 pxq “ ND pxqzt0u. Clearly, if the boundary is smooth at x,
then ND0 pxq “ tανpxq : α ą 0u, where ν : BD Ñ RN denotes the unit outer
normal vector field.
In the following, when dealing with a continuous function f : D Ñ RN ,
defined on a convex body D, with 0 R f pBDq, we denote by degpf, Dq the
Brouwer degree dB pf, int Dq.
Theorem 3. Assume that D Ď RN is a convex body and that f : D Ñ RN
is a continuous function such that
f pxq R ND0 pxq ,
for every x P BD .
(4)
Then, there is an x̄ P D such that f px̄q “ 0. Furthermore, if 0 R f pBDq, then
degpf, Dq “ p´1qN .
5
(5)
Proof. We first notice that it is not restrictive to assume that 0 P int D.
Moreover, we may assume that f pxq ‰ 0 for every x P BD, since otherwise
there is nothing more to prove.
We first assume that D has a smooth boundary, so that ND0 pxq “ tανpxq :
α ą 0u, for every x P BD. Then, we can extend νpxq continuously to the
whole set D; let us still denote by ν : D Ñ RN the vector field thus obtained.
We consider the homotopy F : D ˆ r0, 1s Ñ RN defined by
F px, λq “ p1 ´ λqf pxq ´ λνpxq .
We see that 0 R F pBD ˆ r0, 1sq. Indeed, assume by contradiction that there
are an x P BD and a λ P r0, 1s such that F px, λq “ 0. Then λ ‰ 0, since by
the above assumption f pxq ‰ 0, and λ ‰ 1, since νpxq ‰ 0. So, λ P s0, 1r and,
setting α “ λ{p1 ´ λq, we see that α ą 0, and f pxq “ ανpxq, a contradiction.
Therefore, degpf, Dq “ degp´ν, Dq. On the other hand, since D is convex
and 0 P int D, we have that
xνpxq, xy ą 0 ,
for every x P BD ,
and then degp´ν, Dq “ degp´I, Dq “ p´1qN , from which (5) follows.
We now consider the general case when the boundary of the convex set
D is not necessarily smooth. Then, following [10], for every ε P s0, 1r it
is possible to find a convex body Dε , with a smooth boundary, such that
p1 ´ εqD Ď Dε Ď D, and with the following additional property: denoting
by νε : BDε Ñ RN the unit outer normal vector field, for every p P BDε there
exists some q P BD such that
kp ´ qk ă ε ,
distpνε ppq, ND pqqq ă ε .
So, if ε is small enough, by continuity we will have that
f pxq R tανε pxq : α ą 0u ,
for every x P BDε .
Then, by the first part of the proof, degpf, Dε q “ p´1qN . Since this is true
for every ε ą 0, and since degpf, Dq is well defined, the continuity of the
degree yields (5). The proof is thus completed.
Notice that, when D is a ball centred at the origin, then νpxq “ x{ kxk for
every x P BD, so that conditions (2) and (4) are equivalent. The differences
between these two condition in a general case are illustrated in Figure 1.
Clearly, the avoiding outer cones condition (4) can be replaced by an
avoiding inner cones condition, by just changing the sign of the function f .
6
3 From Poincaré–Bohl to Poincaré–Miranda:
a first generalization
Assumption (1) in the Poincaré–Miranda theorem geometrically means
that either on both faces Fk´ and Fk` the vector field points outwards, or on
both faces it points inwards, for every k “ 1, . . . , N . We will try to replace
this assumption by an avoiding inner/outer cones condition.
We first show how to prove the Poincaré–Miranda theorem by the use of
the Poincaré–Bohl theorem (i.e., by Theorem 2). This will help us towards
some generalizations.
Proof of Theorem 1. We first notice that there is no loss of generality in
assuming that 0 is in the interior of R. Furthermore, we can define a new
function f˜ : R Ñ RN whose components are f˜k “ ˘fk , in such a way that
f˜k px1 , . . . , ak , . . . , xN q ě 0 ě f˜k px1 , . . . , bk , . . . , xN q ,
for every px1 , . . . , xN q P R and every k “ 1, . . . , N . Notice that
f pxq “ 0
ô
f˜pxq “ 0 .
It is easily verified that, for D “ R, the assumptions of Theorem 2 are
satisfied, and the proof is completed.
The above proof shows that, defining an appropriate linear function η :
R Ñ RN with associated matrix being diagonal, and whose elements on the
diagonal are either 1 or ´1, then the transformation f˜ “ η ˝ f satisfies the
assumptions of Theorem 2, and the conclusion immediately follows. With
this idea in mind, we now provide a generalization of Theorem 3.
N
Theorem 4. Let h : RN Ñ RN be a homeomorphism, such that hp0q “ 0,
and assume that D Ď RN is a convex body. Let f : D Ñ RN be a continuous
function such that
f pxq R hpND0 pxqq ,
for every x P BD .
Then, there is an x̄ P D such that f px̄q “ 0.
7
Proof. Define g : D Ñ RN as g “ h´1 ˝ f . Then gpxq R ND0 pxq, for every
x P BD, and Theorem 3 provides the existence of an x̄ P D such that gpx̄q “ 0.
Being h invertible, we have that f px̄q “ 0, as well, thus concluding the
proof.
As a simple and direct consequence of Theorem 4, let N1 and N2 be
positive integers such that N1 ` N2 “ N , and K1 Ď RN1 , K2 Ď RN2 be two
convex bodies. Define, for every x “ px1 , x2 q P K1 ˆ K2 ,
$
if x P BK1 ˆ int K2 ,
& NK1 px1 q ˆ t0u ,
t0u ˆ p´NK2 px2 qq ,
if x P int K1 ˆ BK2 ,
Ac pxq “
%
NK1 px1 q ˆ p´NK2 px2 qq , if x P BK1 ˆ BK2 ,
and let A0c pxq “ Ac pxqzt0u.
Corollary 5. Let f : K1 ˆ K2 Ñ RN be a continuous function such that
f pxq R A0c pxq ,
for every x P BpK1 ˆ K2 q .
(6)
Then, there is an x̄ “ px̄1 , x̄2 q P K1 ˆ K2 for which f px̄q “ 0.
Proof. It is a straightforward application of Theorem 4, with D “ K1 ˆ K2 ,
taking as h the linear transformation defined as the identity IN1 on RN1 and
its opposite ´IN2 on RN2 .
We will refer to the condition (6) as the avoiding cones condition. To compare it with the “classical” condition (1) in the Poincaré–Miranda theorem,
we consider the following example.
Example 6. Let K1 “ ra1 , b1 s and K2 “ ra2 , b2 s, so that D “ R is a rectangle in R2 . We write f pxq “ pf1 pxq, f2 pxqq, and, for simplicity, we assume
that that 0 R f pBDq. Let us denote by x1 a generic point in sa1 , b1 r and
with x2 a generic point in sa2 , b2 r . The comparison between the directions
prohibited by Theorem 1 and those by Corollary 5 is illustrated in Figure 2
and summarized in the following table:
8
D
D
(a) Poincaré–Miranda condition.
(b) Avoiding cones condition.
Figure 2: Comparison between condition (1) of the Poincaré–Miranda theorem and the avoiding cones condition (6). The red halflines
and cones indicate the regions avoided by f pxq for some x P BD.
Poincaré–Miranda
Avoiding cones condition
f1 pa1 , x2 q ă 0
f1 pb1 , x2 q ą 0
f2 px1 , a2 q ą 0
f2 px1 , b2 q ă 0
f1 pa1 , a2 q ă 0 and f2 pa1 , a2 q ą 0
f1 pa1 , b2 q ă 0 and f2 pa1 , b2 q ă 0
f1 pb1 , a2 q ą 0 and f2 pb1 , a2 q ą 0
f1 pb1 , b2 q ą 0 and f2 pb1 , b2 q ă 0
f1 pa1 , x2 q ă 0
f1 pb1 , x2 q ą 0
f2 px1 , a2 q ą 0
f2 px1 , b2 q ă 0
f1 pa1 , a2 q ă 0
f1 pa1 , b2 q ă 0
f1 pb1 , a2 q ą 0
f1 pb1 , b2 q ą 0
or
or
or
or
or
or
or
or
f2 pa1 , x2 q ‰ 0
f2 pb1 , x2 q ‰ 0
f1 px1 , a2 q ‰ 0
f1 px1 , b2 q ‰ 0
f2 pa1 , a2 q ą 0
f2 pa1 , b2 q ă 0
f2 pb1 , a2 q ą 0
f2 pb1 , b2 q ă 0
The same behaviour is observed also in higher dimensions. For a general point
x P BR lying on an pN ´M q-dimensional facet of the rectangle, the Poincaré–
Miranda theorem requires M inequalities, each on a different component of
f pxq. For the same point x, our avoiding cones condition requires much less:
only if all the other N ´ M components of f pxq are null, then at least one of
those M inequalities must be satisfied. This shows that our Corollary 5 also
generalizes [14, Theorem 3.4].
Similar considerations also apply to other variants of the Poincaré–Miranda theorem for sets D which are product of balls instead of intervals, as for
instance in [19, Corollary 2]. For one of these situations, namely the cylinder,
the avoiding cones condition is illustrated in Figure 5.
9
H
C
F
E
D
Figure 3: An example of truncation. The sets C and D are truncated
with respect to F and E is a reconstruction for both.
There is a second way to describe the difference between the avoiding
cones condition (6) and assumption (1) in the Poincaré–Miranda theorem.
Whereas the avoiding cones condition requires that f pxq does not lie in Ac pxq,
the Poincaré–Miranda theorem requires that f pxq actually lies in the polar
cone of Ac pxq, defined as
|c pxq “ tv P RN : xv, wy ď 0 , for every w P Ac pxqu ,
A
besides possibly excluding the trivial case f pxq “ 0.
4 Truncated convex bodies
The Poincaré–Miranda theorem and many of its generalizations consider
a rectangular domain, or at least the product of convex sets. We now want to
replace this structural assumption by introducing a new class of sets, which
will lead us to some topologically different situations.
Given a convex body D and a set F Ď BD, we say that D is truncated
in F if there exists a convex body E and a hyperplane H with the following
properties (see Figure 3):
• F “ D X H, and H is a supporting hyperplane for the set D;
10
• D “ E X HD , where HD is the closed halfspace bounded by H that
includes D;
• the set C :“ EzHD has nonempty interior.
We call E a reconstruction of D with respect to F . Notice that C “ EzD is
a convex body, which is truncated in F , as well.
As possible examples of truncated convex bodies we have rectangles, polytopes and cylinders. Balls, on the contrary, are not truncated. Neither, in
general, having a pN ´ 1q-dimensional face is a sufficient condition to be
truncated: just consider a square with smoothed angles.
In order to investigate the properties of a possible face F , let us denote by
B
F the boundary of F considered as a subset of H. Moreover, along with
normal cones, it is useful to consider also the set-valued analogue of the unit
normal vector ν: it is the map νD , from BD to S N ´1 “ ty P RN : kyk “ 1u,
defined as
*
"
y
0
: y P ND pxq .
νD pxq “
kyk
N ´1
Denoting with conerAs the cone generated by a set A, we then have that
ND pxq “ cone rνD pxqs .
Since the normal cone ND is a map from D to the set of closed, convex subsets
of RN , having closed graph, we can see that νD is an upper semicontinuous
map from BD to the set of compact subsets of RN (for an introduction to
set-valued maps, we refer to [3]).
We remark that our definition (3) of the normal cone for convex sets is
equivalent to setting
(
ND px̄q “ v P RN : xv, x ´ x̄y ď opkx ´ x̄kq , x P D ,
(7)
thus underlining the local nature of the normal cone. This definition is
usually adopted to extend the notion of normal cones to non-convex sets
(see, e.g., [24]).
Proposition 7. If D is a convex body, truncated in F , then
(i) F is closed, convex, and F “ E X H;
11
(ii) F has a non-empty interior if considered as a subset of H;
(iii) if x P B N ´1 F , then νD pxq is multivalued.
Proof. The proof of (i) is immediate, so we start with the proof of (ii). Since
D and C are convex bodies, we can find two open balls BD “ B N ppD , εq Ď D
and BC “ B N ppC , εq Ď C with the same sufficiently small radius ε. We
observe that H separates BD and BC , and so there is a unique point pF in
H X rpD , pC s, the intersection of H with the segment joining pD and pC . We
have
N ´1
ppF , εq “ B N ppF , εq X H Ď E X H Ď F ,
BH
thus showing that F has non-empty interior as a subset of H.
We now prove (iii), so let x P B N ´1 F . First of all we observe that νH P
ND pxq, where νH denotes the unit outward normal vector to HD . Indeed,
F is a convex body with respect to BD, so we can find a sequence pxi qi in
intBD F such that xi Ñ x. Since BD is smooth in intBD F , for every i P N
we have νD pxi q “ νH , and since νD is upper semicontinuous, it follows that
νH P νD pxq.
Let us now consider a sequence pyi qi in BDzF such that yi Ñ x, and let
us pick vi P νD pyi q. Then, by compactness, there exist a subsequence pvik qk
and a vector νv P νD pxq such that vik Ñ νv . We claim that νv ‰ νH ; from
this it follows that νD pxq has at least two elements (and so actually, by the
convexity of ND pxq, infinite elements). Taking p P EzD, we have
xνH , p ´ xy ą 0 .
On the other hand, since ND pyik q “ NE pyik q because of (7), we have
0 ě xvik , p ´ yik y Ñ xνv , p ´ xy ,
and so xνv , p ´ xy ă xνH , p ´ xy, thus showing that νv ‰ νH .
An immediate consequence is that smooth convex bodies are not truncated. We can interpret (iii) as the necessity for D to have “edges” on the
boundary of F .
We now consider multiple truncations. Given a convex body D Ď RN
and a family tF1 , . . . , FM u of pairwise disjoint sets, we say that D is truncated in tF1 , . . . , FM u if there exists a convex body E and some hyperplanes
H1 , . . . , HM with the following properties:
12
• for every i, Fi “ D X Hi , and Hi is a supporting hyperplane for the set
D;
1
M
i
• D “ E X HD
X ¨ ¨ ¨ X HD
, where HD
is the closed halfspace bounded
by Hi that includes D;
i
has nonempty interior.
• for every i, the set Ci :“ EzHD
We call E a reconstruction of D with respect to tF1 , . . . , FM u. Notice that
each Ci is a convex body, which is truncated in Fi . Moreover, the sets Ci are
pairwise disjoint, and one has
C1 Y ¨ ¨ ¨ Y CM “ EzD .
Example 8 (Polygons and polyhedra). In R2 , a polygon with faces Fj is
truncated in tF1 , . . . , FM u if the faces F1 , . . . , FM are not pairwise adjacent.
The simplest way to construct a convex body truncated in M faces is to
consider the 2M -agon as truncated on alternate faces.
For polyhedra in R3 we need that the faces where truncations occur do not
share any vertices. Thus, the cube can be truncated in at most two (opposite)
faces, and so the octahedron, while the icosahedron can be truncated in at
most four faces. One way to construct polyhedra truncated in M faces is to
consider the prism with a 2M -agonal base as truncated on alternate lateral
faces.
5 Optimal reconstructions
Let us spend a few words about reconstructions. Clearly, for every truncated convex body D, there are infinitely many possible reconstructions; our
plan is to focus on some special reconstructions which are optimal for our
purposes. They will indeed minimize the cones Ac pxq to be avoided by the
vector field, and hence provide the best choice for the application of the
results to be stated in Section 6. Some preliminary remarks are in order.
Given a point x̄ P BD, to every vector v P ND0 px̄q we can associate the
supporting hyperplane containing x̄,
Hv “ tx̄ ` w : xv, wy “ 0u ,
13
and the corresponding halfspace containing D:
Hv “ tx P RN : xv, x ´ x̄y ď 0u .
Being D a convex body, it coincides with the intersection of its supporting
halfspaces [12, Prop. 2 p. 58].
Given x̄ P BD, we can consider the intersection of all those supporting
halfspaces whose boundary contains x̄. Using the relationship with the normal cone, we can write this intersection as
qD px̄q .
tx P RN : xv, x ´ x̄y ď 0 , for every v P ND px̄qu “ x̄ ` N
(8)
qD px̄q of the normal cone is the so-called tangent cone [5, 24].
The polar N
In the following, we denote by convrAs the convex hull of a given set A,
that is the smallest convex set including A. The following lemma is a first
step towards optimal reconstructions.
Lemma 9. Let D Ď RN be a convex body truncated in F . Then, there exists
a closed convex set Emax such that Emax X HD “ D, with the property that,
if E is any reconstruction of D with respect to F , then E Ď Emax .
Proof. If x P BDzF , then, in a sufficiently small neighbourhood, of x, the
set D coincides with any reconstruction E with respect to F , and hence
ND pxq “ NE pxq. This means that E and D have the same supporting
qD pxq.
hyperplanes containing x and therefore, by (8), we have that E Ď x ` N
Now, let us set
ı
č ”
qD pxq .
x`N
Emax “
xPBDzF
By what we have just seen, it follows that E Ď Emax for every possible
reconstruction E. Hence, D Ď Emax X HD . Furthermore, Emax is a closed
convex set since it is the intersection of closed convex sets.
We want to prove that Emax X HD “ D. First of all, we prove that
BD Ď BpEmax X HD q. Indeed, each point x of BD belongs to Emax X HD , since
D Ď Emax X HD . If x P BDzF , then there is a supporting hyperplane of Emax
containing x; on the other hand, if x P F , then x P HD . In any case, there is
a supporting hyperplane of Emax X HD containing x, so x P BpEmax X HD q.
14
Suppose now by contradiction that there exists y P Emax X HD such that
y R D. Let U “ Bpx0 , rq be an open ball contained in D. By convexity, there
exists a unique x̄ P BD X rx0 , ys. It is easy to show that there exists an open
neighbourhood V of x̄ such that V Ď convrU Y tyus Ď Emax X HD . Then,
x̄ R BpEmax XHD q, contradicting the fact that x̄ P BD. Thus, Emax XHD “ D,
and the proof is completed.
An immediate consequence of the above lemma is that Emax is the smallest
set containing every reconstruction of D with respect to F . More precisely,
since the intersection of Emax with any arbitrarily large closed ball containing
D is a reconstruction, we deduce that every point of Emax is contained in a
reconstruction. So, Emax is the union of all possible reconstructions of D
with respect to F .
We say that a reconstruction E is optimal if, for every x P F ,
NC pxq “ NCmax pxq ,
where Cmax “ Emax zD .
Since, for every reconstruction, the inclusion NC pxq Ě NCmax pxq holds for
every x P F , an optimal reconstruction minimizes NC pxq, as illustrated in
Figure 4.
In general, Emax is not bounded and therefore it is not a reconstruction;
however it is always possible to build an optimal reconstruction simply taking
E “ Emax X K, where K is a convex body such that D Ď int K. Moreover,
one can find an optimal reconstruction E which is as close to D as desired.
Indeed, given ε ą 0, it suffices to take E “ Emax X BrD, εs, where
BrD, εs “ tx P RN : distpx, Dq ď εu ,
to have an optimal reconstruction whose distance from D is at most ε.
Example 10 (Cylinders/prisms). Let D “ K ˆ r´1, 1s, where K is a convex
body in RN ´1 . Then, D is truncated in any of its two bases. For instance,
we can take H “ RN ´1 ˆ t1u, and F “ K ˆ t1u. In this case, we see that
Emax “ K ˆ r´1, `8q, and a possible optimal reconstruction with respect
to the face F is given by E “ D Y C, where C “ K ˆ r1, 2s. Notice that, if
instead of C we take, e.g., C 1 “ tpx, y ` 1q : x P K, 0 ď y ď distpx, BKqu, it
is true that we have a reconstruction, but it is not optimal.
15
Cmax
C
F
D
Figure 4: An example of non-optimal reconstruction E “ D Y C
(blue), compared with the optimal reconstruction Emax (red, starred),
with emphasis on their respective normal cones.
Example 11 (Polytopes). If D is a convex polytope with faces F1 , . . . , Fm ,
it is truncated with respect to any of them. Let us focus on a particular one,
F “ Fj . Correspondingly, we will have
Ej,max “
m
č
i
HD
,
i“1
i‰j
i
where HD
denotes the halfspace including D bounded by the supporting
hyperplane Hi generated by the face Fi . If Ej,max is bounded, then it is an
optimal reconstruction. Concerning Cj , it is a triangle if N “ 2, while, if
N ě 3, it is the polytope with Fj as a face, whereas the other faces are given
by the prolongations of the faces of D adjacent to Fj . This is the case, for
instance, of the regular m-agon for m ě 5 in R2 , of the dodecahedron and
of the icosahedron in R3 . The set Ej,max is unbounded, for instance, when
dealing with regular simplices, where Ej,max is the cone generated by Fj and
the vertex of the simplex opposed to Fj . (Squares, cubes and hypercubes
have unbounded Ej,max , but have been already discussed in the previous
16
example.) Clearly, non-regular polytopes can have at the same time faces
with bounded and faces with unbounded Ej,max .
In the case of a convex body D truncated at tF1 , . . . , FM u, we say that a
reconstruction E “ D Y C1 Y ¨ ¨ ¨ Y CM is optimal if, for every truncation Fi ,
the reconstruction Ei “ D Y Ci is optimal.
6 Main results
Let D Ď RN be a convex body truncated in tF1 , . . . , FM u, with an optimal
reconstruction E “ D Y C1 Y ¨ ¨ ¨ Y CM . We define the map Ac , from BD to
the closed, convex cones of RN , as
#
NCi pxq , if x P Fi ,
Ac pxq “
Ť
ND pxq , if x P BDz M
i“1 Fi .
Moreover, as usual, we denote by A0c pxq the set Ac pxq deprived of the origin.
We now state the main theorem of this paper.
Theorem 12. Let D Ď RN be a convex body truncated in tF1 , . . . , FM u, with
M ě 2, and let f : D Ñ RN be a continuous function satisfying
f pxq R A0c pxq , for every x P BD .
Then, there is an x̄ P D such that f px̄q “ 0. Furthermore, if 0 R f pBDq, then
degpf, Dq “ p´1qN p1 ´ M q.
The proof of Theorem 12 will be given in Section 8. We now provide
some examples where it can be applied.
Example 13 (Cylinders/prisms). Let D “ K ˆ r´1, 1s, where K Ď RN ´1 is
a convex body. The set D is truncated in F´ “ K ˆ t´1u and F` “ K ˆ t1u,
and we have
$
ND pxq ,
if x P BKˆ s ´ 1, 1r ,
’
’
’
&´N pxq ,
if x P int K ˆ t´1, 1u ,
D
Ac pxq “
’
NK pyqˆ s ´ 8, 0s , if x “ py, 1q , with y P BK ,
’
’
%
NK pyq ˆ r0, `8r , if x “ py, ´1q , with y P BK .
17
D
Figure 5: Avoiding cones in the case of the cylinder.
These cones are illustrated for the three-dimensional case in Figure 5, where
K is a circle in R2 . We remark that, in the case of cylinders, Theorem 12
coincides with Corollary 5.
Example 14 (Polytopes). Let the convex polytope D, with faces Fj , be
truncated in tF1 , . . . , FM u. For every x P BD, we denote by Ipxq “ ti : x P Fi u
the set of indices of those faces containing x, and by νi the outward unit vector
normal to Fi . Furthermore, we denote by σpiq the sign of the avoiding cones
condition in Fi , namely
#
´1 , if i “ 1, . . . , M (avoiding inner normal cones) ,
σpiq “
`1 , otherwise (avoiding outer normal cones) .
Then, Ac pxq corresponds to the convex cone generated by the set
tσpiqνi : i P Ipxqu ,
whose elements are the outer/inner normal cones assigned by Ac to the points
in the interior of the faces containing x. We illustrate in Figure 6 the particular case of an hexagon truncated in three alternate faces. We highlight
that, being in this case N “ 2 and M “ 3, if f satisfies the avoiding cones
condition of Theorem 12, then
degpf, Dq “ p´1q2 p1 ´ 3q “ ´2 .
18
D
Figure 6: Avoiding cones in the case of a hexagon.
We finally notice that Theorem 3 can be interpreted as a version of Theorem 12, with M “ 0. So, having Theorem 4 in mind, we can also write the
following extension of Theorem 12.
Theorem 15. Let h : RN Ñ RN be a homeomorphism, such that hp0q “ 0,
and assume that D Ď RN is a convex body truncated in tF1 , . . . , FM u, with
M ě 2. Let f : D Ñ RN be a continuous function such that
f pxq R hpA0c pxqq ,
for every x P BD .
Then, there is an x̄ P D such that f px̄q “ 0.
Until now, the domain D of our functions has been supposed to be a
convex body. However, all our results can be easily extended to sets D which
are just diffeomorphic to a convex body D. By this we mean that there are
two open sets A, B in RN , with D Ď A, D Ď B, and a diffeomorphism
ϕ : A Ñ B, such that
D “ ϕpDq .
To define the normal cone to D at a boundary point y P BD, let ψ “ ϕ´1 :
B Ñ A, so that ψpyq P BD, and set
ND pyq “ pψ 1 pyqqT ND pψpyqq .
19
0
D
D
x1
x1
x2
(a) Poincaré–Bohl condition (2).
x2
(b) Avoiding outer cones condition (9).
Figure 7: Comparison between condition (2) of the Poincaré–Bohl
theorem and the avoiding outer cones condition (9). The red halflines
and cones indicate the regions avoided by f pxq for some x P BD.
We remark that this choice preserves the extended notion of normal cone
recalled in (7).
Let us first go back to our variant of the Poincaré–Bohl theorem. Writing,
as usual, ND0 pyq “ ND pyqzt0u, we have the following extension of Theorem 3.
Theorem 16. Assume that D is a subset of RN , diffeomorphic to a convex
body. Let f : D Ñ RN be a continuous function such that
f pyq R ND0 pyq ,
for every y P BD .
(9)
Then, there is a ȳ P D such that f pȳq “ 0.
Proof. Using the above notation, we have D “ ψpDq and we define f˜ : D Ñ
RN as
f˜pxq “ pϕ1 pxqqT f pϕpxqq .
Then, condition (4) holds replacing f by f˜, so Theorem 3 applies, and we
easily conclude.
In Figure 7 we illustrate the avoiding cones condition of Theorem 16, in
the case when D has a smooth boundary.
20
Now, in order to extend Theorem 12, let us consider a set D which is
diffeomorphic to a convex body D, truncated in tF1 , . . . , FM u. Since D Ď A,
D Ď B and both sets A an B are open, we can choose a reconstruction E
of D with respect to tF1 , . . . , FM u, even an optimal reconstruction, to be
contained in A, as well. Setting
E “ ϕpEq , F1 “ ϕpF1 q , . . . , FM “ ϕpFM q ,
we say that E is a reconstruction of D with respect to tF1 , . . . , FM u. We
also say that D is truncated in tF1 , . . . , FM u. Then, referring to the notation
introduced in Section 4, we have E “ D Y C1 Y ¨ ¨ ¨ Y CM , and setting
C1 “ ϕpC1 q, . . . , CM “ ϕpCM q, we can define the cones
#
NCi pyq , if y P Fi ,
Ac pxq “
Ť
ND pyq , if y P BDz M
i“1 Fi .
Writing, as usual, A0c pxq “ Ac pxqzt0u, we can state the following.
Theorem 17. Let D Ď RN , diffeomorphic to a convex body, be truncated in
tF1 , . . . , FM u, and let f : D Ñ RN be a continuous function satisfying
f pyq R A0c pyq , for every y P BD .
Then, there is a ȳ P D such that f pȳq “ 0.
An example of the avoiding cones condition of Theorem 17 is illustrated
in Figure 8, where the set D is diffeomorphic to a hexagon D (cf. Figure 6).
We end this section with the analogue of Theorem 15.
Theorem 18. Let h : RN Ñ RN be a homeomorphism, such that hp0q “
0, assume that D Ď RN , diffeomorphic to a convex body, is truncated in
tF1 , . . . , FM u, and let f : D Ñ RN be a continuous function satisfying
f pyq R hpA0c pyqq , for every y P BD .
Then, there is a ȳ P D such that f pȳq “ 0.
21
D
Figure 8: Avoiding cones in the case of a non-convex set diffeomorphic to a hexagon.
7 An application to multiple saddles
In this section we will show that our results can be applied to deal with
the gradient of a potential V having degenerate multiple saddle points, where
multiple expansive and contractive directions appear.
A detailed exposition for the planar case can be found in [7], where the
authors considered k-fold saddles formed by the alternation, around the critical point, of k ` 1 ascending directions and k ` 1 descending directions:
the first ones identified by trajectories of the flow of ∇V escaping from the
critical point, while the second ones by trajectories converging to the critical
point. (For a similar situation, see also [1, 8, 9].) With this description, the
standard non-degenerate saddle is an example of 1-fold saddle, whereas the
monkey saddle is a 2-fold saddle (cf. Figure 9).
In higher dimensions, the criterion of alternation is no longer applicable
and more sophisticated situations may arise. Different approaches, mainly
related to the Conley index or some of its generalizations, have been used
to study the degree in the case of higher dimensional multiple saddle points
22
Figure 9: The monkey saddle V px1 , x2 q “ x31 ´ 3x1 x22 .
(cf. [6, 25, 28]). We propose here a simpler strategy, based on our Theorem 12,
to recover some of those results.
We consider a continuously differentiable function V : Br0, Rs Ñ R, and
assume that, near the boundary of the domain, namely for 0 ă r ď kxk ď R,
it can be written in the form
˙
ˆ
x
,
(10)
V pxq “ ρpkxkqS
kxk
where ρ : rr, Rs Ñ s0, `8r and S : S N ´1 Ñ R are continuously differentiable
functions, and ρ1 pξq ą 0, for every ξ P rr, Rs. This factorization, in a certain
sense, generalizes the idea of positive homogeneity, which corresponds to the
choice ρptq “ tα , for a certain α ą 0.
In this region of the domain, ∇V pxq can be decomposed in radial and
tangential components as
ˆ
˙
ˆ
˙
x
1
x
x
1
∇V pxq “ ρ p|x|q
S
`
∇S S
,
(11)
kxk
kxk
kxk
kxk
where ∇S Spxq denotes the tangential gradient of Spxq, i.e.,
ˆ
˙ B
ˆ
˙F
x
x
∇S Spxq “ ∇S
´ x, ∇S
x.
kxk
kxk
We see that ∇S S corresponds to the surface gradient on the unit sphere of
the function x ÞÑ Spx{ kxkq, defined on RN zt0u.
23
Since in (11) the two terms in the sum are orthogonal, their sum vanishes
if and only if they are both zero. Hence, if r ď kxk ď R, we have that
˙
ˆ
˙
ˆ
x
x
“ 0 and ∇S S
“ 0.
∇V pxq “ 0 ô S
kxk
kxk
In particular, if we want the degree degp∇V, Br0, Rsq to be well defined,
we need to ask that Spxq and ∇S Spxq do not vanish simultaneously at any
x P S N ´1 .
Let us state the main result of this section.
Theorem 19. In the above setting, assume that
(i) if x P S N ´1 satisfies ∇S Spxq “ 0, then Spxq ‰ 0;
(ii) the set tx P S N ´1 : Spxq ě 0u is the union of M disjoint subsets, which
are diffeomorphic to an pN ´ 1q-dimensional ball.
Then, degp∇V, Br0, Rsq “ p´1qN p1 ´ M q.
Proof. If M “ 0, we have that Spxq ă 0 for every x P S N ´1 . Taking
D “ Br0, Rs, we have that, for every x P BD, the cone to avoid is ND pxq “
tλx : λ ě 0u. Being V pxq ă 0, the radial component of ∇V pxq is not zero
and points inward, so ∇V pxq R ND pxq. Theorem 3 can then be applied to
conclude.
So, from now on, we can assume M ě 1. Given a vector y P S N ´1 , for
every x P S N ´1 such that x ‰ ˘y, we define
σpx; yq “
y ´ x ´ xx, y ´ xy x
.
ky ´ x ´ xx, y ´ xy xk
It is the unit vector on the tangent space to S N ´1 in x, associated to the
shortest path from x to y. We say that a local maximum point y P S N ´1 for
S is regular if there exists a neighbourhood U of y, with the property that
xσpx; yq, ∇S Spxqy ą 0 , for every x P U X S N ´1 .
This condition is true, for instance, if y is a non-degenerate local maximum
point. We first prove the theorem when (ii) is replaced by the following
stronger assumption:
24
(ii˚ ) if y P S N ´1 satisfies ∇S Spyq “ 0 and Spyq ě 0, then y is a regular local
maximum point for S, and Spyq ą 0. Moreover, there are exactly M
of such points.
Let s1 , . . . , sM be the regular maximum points of condition (ii˚ ). For any
ε P s0, R ´ rr , we set
Hi “ tx P RN : xx, si y ď R ´ εu .
Let
D “ Br0, Rs X H1 X H2 X ¨ ¨ ¨ X HM ,
and define Hi “ BHi . If ε is sufficiently small, then D is a convex body
truncated in tF1 , . . . FM u, with Fi “ Br0, Rs Y Hi , and E “ Br0, Rs is an
optimal reconstruction. Let us verify that the avoiding cones condition holds,
provided that ε is sufficiently small.
Ť
For x P BDz M
i“1 Fi , the cone to avoid is Ac pxq “ ND pxq “ tλx : λ ě 0u.
If V pxq ă 0, then the radial component of ∇V pxq is not zero and points
inward, so ∇V pxq R Ac pxq. If V pxq ě 0, since x ‰ Rsi for each i “ 1, . . . , M ,
the tangential component of ∇V pxq is not zero and so ∇V pxq R Ac pxq.
If x P intBD Fi , for some i “ 1, . . . , M , then Ac pxq “at´λsi : λ ě 0u. (Note
that intBD Fi is a pN ´ 1q-dimensional ball of radius εp2R ´ εq centred in
pR ´ εqsi .) Since ∇V ppR ´ εqsi q “ λi si , for some λi ą 0, if ε is sufficiently
small, by continuity, it has to be ∇V pxq R Ac pxq.
If x P B N ´1 Fi , the boundary with respect to Hi , for some i “ 1, . . . , M ,
then Ac pxq is the convex cone generated by
? t´si , xu. By definition, we have
that xσpx; si q, xy “ 0 and, if kx ´ Rsi k ď 2R, then also xσpx; si q, ´si y ď 0.
Thus, if ε is sufficiently small, xσpx; si q, vy ď 0 for every v P Ac pxq. On the
other hand, since si is a regular maximum point, taking ε sufficiently small
we get
ˆ
˙F
B
x
1
xσpx; si q, ∇V pxqy “ σpx; si q,
∇S S
ą 0,
kxk
kxk
so that ∇V pxq R Ac pxq.
So, in all cases, we have that ∇V pxq R Ac pxq. Then, by Theorem 12,
degp∇V, Dq “ p´1qN p1 ´ M q. Since there are no critical points of V in
Br0, RszD, the excision property of the degree leads us to the end of the
proof, in this case.
25
Let us now consider the general case. We write
tx P S N ´1 : Spxq ě 0u “ Σ1 Y ¨ ¨ ¨ Y ΣM ,
and assume that, for every i “ 1, . . . , M , there are an open set Ui containing
Σi , an open set Vi containing Br0, 1s and a diffeomorphism ψi : Ui Ñ Vi ,
such that ψi pΣi q “ Br0, 1s; moreover, the sets Ui can be assumed pairwise
disjoint.
Define Pi : Ui Ñ R as
Pi pxq “ 1 ´ }ψi pxq}2 .
Then, for every x P BΣi , there is a λi pxq ą 0 for which ∇Spxq “ λi pxq∇Pi pxq.
Hence, for δ ą 0 sufficiently small, Br0, 1 ` δs Ď Vi and, writing Uiδ “
ψi´1 pBr0, 1 ` δsq, we have that Σi Ď Uiδ Ď Ui . Furthermore, for δ sufficiently
small, we have also
x∇Spxq, ∇Pi pxqy ą 0 , for every x P Uiδ zΣi .
Let µ : R Ñ R be an increasing continuously differentiable function such
that
"
0 , if s ď 0 ,
µpsq “
µ1 p0q “ µ1 pδq “ 0 .
1 , if s ě δ ,
Define W : S 1 ˆ r0, 1s Ñ R as follows:
´
¯ı
$ ”
’
1
´
µ
distpψpxq,
Br0,
1sq
pλPi pxq ` p1 ´ λqSpxqq`
’
’
´
¯
&
`µ distpψpxq, Br0, 1sq Spxq , if x P Uiδ , for some i ,
W px, λq “
’
’
’
%
Spxq ,
otherwise .
This function is continuously differentiable and transforms Spxq “ W px, 0q
r
into a function Spxq
“ W px, 1q, satisfying (ii˚ ). Moreover, the following two
additional properties hold:
- the sign of W px, λq does not depend on λ;
- the functions W p¨, λq have no critical points y with W py, λq “ 0.
26
Such a function W induces an admissible homotopy H : Br0, Rs ˆ r0, 1s Ñ
RN , defined as
„
ˆ
˙
x
Hpx, λq “ ∇ ρpkxkq W
,λ
,
kxk
which transforms ∇V pxq “ Hpx, 0q into ∇Vr pxq, where Vr pxq satisfies the
assumptions of the theorem, and also the additional condition (ii˚ ). Since
the admissible homotopy preserves the degree, the proof is completed.
The following symmetrical version of Theorem 19 holds.
Theorem 20. Let the assumptions of Theorem 19 hold, with only (ii) replaced by
(ii´ ) the set tx P S N ´1 : Spxq ď 0u is the union of M disjoint subsets, which
are diffeomorphic to an pN ´ 1q-dimensional ball.
Then, degp∇V, Br0, Rsq “ 1 ´ M .
Proof. It is sufficient to apply Theorem 19 to ´V instead of V .
The above result should be compared with [28, Theorem 4.4], which is
stated in a more general setting. We also notice that, when M “ 0, Theorem 20 is related to a result by Krasnosel’skii [15] (see also [2]) stating that,
when V is coercive, then, for R large enough, degp∇V, Br0, Rsq “ 1 .
In the planar case, conditions (ii) and (ii´ ) can be simplified, as follows.
Corollary 21. Let the assumptions of Theorem 19 hold, for N “ 2, with
only (ii) replaced by
(ii2 ) the function S changes sign exactly 2M times on S 1 .
Then, degp∇V, Br0, Rsq “ 1 ´ M .
Proof. Since the zeros of S are simple, the set tx P S 1 : Spxq ě 0u is the
union of M disjoint arcs, each of which is diffeomorphic to a compact interval
of R.
We have thus recovered, in the planar case, a variant of the alternation
criterion described in [7]. We now give two simple examples where our results
directly apply. The first one deals with a planar situation.
27
Example 22. Let us consider, for a positive integer k, the family of potentials
Sk psq “ cosrpk ` 1qss ,
where s P r0, 2πr is the angle which determines a point x P S 1 . Taking
ρk ptq “ tk`1 and identifying R2 with the complex plane, we get
Vk pzq “ ρk p|z|qSk parg zq “ <pz k`1 q .
The saddle generated by Sk has k ` 1 ascending directions at the points of
maximum for Sk , namely s “ 2jπ{pk ` 1q, with j “ 0, 1, . . . , k, and k ` 1
descending directions at the points of minimum for Sk , i.e., s “ p2j `1qπ{pk`
1q. We thus see that this choice of Sk produces a model of k-fold saddle for
every k ě 1. In this case, degp∇Vk , Br0, Rsq “ ´k, for any R ą 0.
As we said above, our main purpose is to study also non-planar situations.
In our second example we show an illustrative application in R3 .
Example 23. Let v1 , v2 , v3 , v4 be the vertices of a tetrahedron centred in the
origin, namely
? ˙
? ˙
ˆ ?
ˆ
6
3
1
6
, v2 “ ´
,´ ,´
,
v1 “ 0, 0,
4
6
2
12
? ˙
? ˙
ˆ ?
ˆ?
3 1
6
3
6
v3 “ ´
, ,´
, v4 “
, 0, ´
.
6 2
12
3
12
Let us consider the functions Va , Vb : R3 Ñ R, defined as
«
ˆ
˙2 ff
x
1
Va pxq “ kxk2
´ min dist
, vi
,
5 i“1,...,4
kxk
Vb pxq “
4
ź
i“1
xx, vi y ´
kxk4
.
150
Both potentials admit the factorization (10), since they are positively homogeneous of degree two and four, respectively. The behaviour of their spherical
components Sa pxq and Sb pxq is illustrated in Figure 10.
The potential Sa has four positive maximum points, placed in correspondence of the vertices of the tetrahedron, four negative minima, in correspondence of the centers of the faces of the tetrahedron, and six negative saddle
points, in correspondence of the midpoints of the edges of the tetrahedron.
28
(a) The potential Sa pxq.
(b) The potential Sb pxq.
Figure 10: Behaviour of the functions Sa psq and Sb psq of Example 23
on the unit sphere. The black thick line indicates where they take
value zero. The functions are positive in the red regions and negative
in the blue ones.
The potential Sb instead has six positive maximum points, placed in correspondence of the midpoints of the edges of the tetrahedron, defining in
this way the vertices of an octahedron. It also has eight negative minima, in
correspondence of both the vertices and the centers of the faces of the tetrahedron (i.e., the centers of the faces of the octahedron), and twelve negative
saddle points, corresponding to the midpoints of the edges of the octahedron.
Moreover, we observe that both Va and Vb satisfy the hypotheses of Theorem 19, with Ma “ 4 and Mb “ 6, respectively, so that, for every R ą 0, we
have
degp∇Va , Br0, Rsq “ p´1q3 p1 ´ Ma q “ 3 ,
degp∇Vb , Br0, Rsq “ p´1q3 p1 ´ Mb q “ 5 .
8 Proof of Theorem 12
In this section, in order to provide a proof for Theorem 12, we will need
some basic facts from the theory of set-valued maps, for which we refer to
the book of Aubin and Cellina [3].
29
We denote by πD : RN Ñ D the projection on the convex set D: for every
x̄ P RN , πD px̄q is the only element of D satisfying
distpx̄, πD px̄qq ď distpx̄, xq ,
for every x P D .
We remark that
πD px̄ ` vq “ x̄
v P ND px̄q .
ô
Let us start showing that if D is a convex body, then, for every x P D,
v P ND0 pxq
´v R ND0 pxq .
ñ
Indeed, if on the contrary both v and ´v belong to ND0 pxq, then, for every
x P D, it would be
0 ě xv, x ´ x̄y “ ´ x´v, x ´ x̄y ě 0 .
Hence, D would be included in a hyperplane orthogonal to v and so it would
have empty interior, in contradiction with the assumption of being a convex
body.
The following lemma will be crucial for the proof of Theorem 12.
Lemma 24. Let D Ď RN be a convex body truncated in F , and E “ D Y C
be a reconstruction of D with respect to F . If f : D Ñ RN is a continuous
map such that f pxq R NC pxq, for every x P F , then f can be extended to a
continuous function fˆ: E Ñ RN , such that
fˆpxq R NC pxq ,
for every x P BC .
pC from BC
Proof. The core of the proof is to show the existence of a map N
N
to the closed, convex cones of R , with closed graph and such that
pC pxq and
(N1) for every x P BC, NC pxq Ď N
pC pxqzt0u
vPN
ñ
pC pxq ;
´v R N
pC admits a continuous selection α : BC Ñ RN such that
(N2) N
pC zt0u ,
αpxq P N
pC pxq, for every x P F .
(N3) f pxq R N
30
for every x P BC ;
Step 1. Let us define the set-valued map Φ from BC to RN as
Φpxq “ conv rνC pxqs .
Its values are convex and compact. Let us show that Φ is upper semicontinuous. To do so, we first observe that, for a compact convex set K Ď RN , the
ε-neighbourhood BpK, εq is convex because of the convexity of the euclidean
distance. Now, take x P BC and fix ε ą 0. Since νC is upper semicontinuous
and BpΦpxq, εq is a neighbourhood of νC pxq, there exists a neighbourhood U
of x in BC such that νC pU q Ď BpΦpxq, εq. From the convexity of BpΦpxq, εq,
it follows that ΦpU q Ď BpΦpxq, εq. The upper semicontinuity of Φ is thus
proved.
Since Φpxq Ď NC pxq, we have that
v P Φpxqzt0u
ñ
´v R Φpxq .
Let us now prove that 0 R ΦpBCq. Suppose by contradiction that 0 P Φpxq
for some x P BC; then there exist v1 , . . . , vk P νC pxq and λ1 , . . . , λk in s0, 1r ,
with λ1 ` ¨ ¨ ¨ ` λk “ 1, such that
0“
k
ÿ
λi vi “ λ1 v1 ` p1 ´ λ1 qṽ ,
i“1
k
ÿ
λi v i
P Φpxq .
with ṽ “
1 ´ λ1
i“2
Let us set µ “ mintλ1 {2, p1 ´ λ1 q{2u; then,
0 ‰ w1 “ pλ1 ` µqv1 ` p1 ´ λ1 ´ µqṽ P Φpxq ,
0 ‰ w2 “ pλ1 ´ µqv1 ` p1 ´ λ1 ` µqṽ P Φpxq ,
and so w2 “ ´w1 , in contradiction with the fact that Φpxq does not contain
opposite vectors. Hence, 0 R Φpxq for every x P BC.
Since Φ is upper semicontinuous and thus has a closed graph, we can set
δ0 :“ distpBC ˆ t0u, graph Φq ą 0 .
(12)
BC
Furthermore, we note that ΦpBCq Ď Br0, 1s, for the convexity of the euclidean
distance, and so ΦpBCq is compact.
31
Step 2. Since 0 R f pF q, we can define f1 : F Ñ S N ´1 Ď RN as
f1 pxq “
f pxq
.
kf pxqk
The function f1 is continuous and the hypothesis f pxq R NC pxq is equivalent
to f1 pxq R νpxq, from which it follows that f1 pxq R Φpxq, for every x P F .
Thus we can define
δ1 :“ distpgraph f1 , graph Φq ą 0 .
F
(13)
BC
We remark that we are considering the distance in RN ˆ RN between two
compact sets corresponding to the graphs of two functions with different
domains.
By [3, Sect. 1.13, Theorem 1] (cf. also [13]), there exists a sequence of
upper semicontinuous set-valued maps Φi , from BC to RN , satisfying
(S1) for every i P N, Φi has a continuous selection αi ;
(S2) for every i P N, Φi has closed graph and compact values;
(S3) for every x P BC,
Φpxq Ď ¨ ¨ ¨ Ď Φi`1 pxq Ď Φi pxq Ď ¨ ¨ ¨ Ď Φ0 pxq ,
and
Φpxq “
č
Φi pxq .
iPN
Moreover, since ΦpBCq is compact, the maps Φi can be taken with convex
values.
Let us introduce the set-valued maps νi from BC to RN as
"
*
y
: y P Φi pxqzt0u .
νi pxq “
kyk
Note that the maps νi have compact graph. Moreover, for every x P BC,
č
νi`1 pxq Ď νi pxq , and
νi pxq “ νC pxq .
iPN
32
From this and the continuity of the distance, we get that there exists an
index i1 P N such that, for every i ě i1 ,
distpgraph f1 , graph νi q ą
F
BC
δ1
,
2
(14)
where δ1 has been defined in (13). Similarly, from (12) we get that there
exists ı̄ ě i1 such that 0 R Φi pBCq, for every i ě ı̄.
Step 3. We claim that, for any j ě ı̄, the choice
pC pxq “ conerΦj pxqs
N
satisfies all the requirements (N1), (N2) and (N3). First of all we notice that
the cone generated by a compact, convex set is always closed and convex.
pC
Similarly, since the graph of Φj is compact, it follows that the graph of N
pC pxq, it follows that
is closed. Furthermore, since νC pxq Ď Φpxq Ď Φj pxq Ď N
pC pxq.
NC pxq Ď N
Now let us suppose by contradiction that, for some x P BC, there exists
pC pxqzt0u such that ´v P N
pC pxq. Then there exist v1 “ a1 v and
v P N
v2 “ ´a2 v, with a1 ą 0, a2 ą 0, such that both v1 P Φj pxq and v2 P Φj pxq.
Since Φj pxq is convex, it follows that
0“
a1
a2
v1 `
v2 P Φj pxq ,
a1 ` a2
a1 ` a2
in contradiction with j ě ı̄. Hence, (N1) is satisfied.
To satisfy (N2) it is sufficient to take α “ αj , where αj is a continuous
selection of Φj given by (S1). Since 0 R Φj pxq for every x P BC, we have that
αj pxq ‰ 0 for every x P BC.
Let us now define ν̂C pxq “ νj pxq, for a fixed j ě ı̄. Then, from (14) we
have the estimate
δ1
distpgraph f1 , graph ν̂C q ą ,
2
F
BC
from which (N3) follows straightforwardly.
33
Step 4. Now we are ready to construct the sought prolongation fˆ. Let us
pick any 0 ă δ ă δ1 {2. We define Fδ “ BC X BpF, δq and introduce the
function f2 : Fδ Ñ S N ´1 Ď RN as
f2 pxq “ f1 pπF pxqq “
f pπF pxqq
.
kf pπF pxqqk
For every x P Fδ we have
`
˘
`
˘
dist px, f2 pxqq, graph f1 ď dist px, f2 pxqq, pπF pxq, f2 pxqq ď δ .
F
Using the triangle inequality, this implies
distppx, f2 pxqq, graph ν̂C q ě
BC
ě distpgraph f1 , graph ν̂C q ´ distppx, f2 pxqq, graph f1 q
F
ě
F
BC
δ1
´ δ ą 0,
2
and so
distpgraph f2 , graph ν̂C q ě
Fδ
BC
δ1
´ δ ą 0,
2
from which it follows that f2 pxq R ν̂pxq, for every x P Fδ , and consequently
pC pxq.
f pπF pxqq R N
Writing λx “ distpx, F q{δ, we now set
$
’
if x P D ,
&f pxq ,
ˆ
f pxq “ p1 ´ λx qf pπF pxqq ´ λx αpxq , if x P Fδ zF ,
’
%
´αpxq ,
if x P BCzFδ ,
where α : BC Ñ RN is the continuous selection provided by (N2). We thus
have a continuous function defined on D Y BC. If we prove that fˆ satisfies
the desired property on BC, then the proof is completed, since we can apply
Tietze’s theorem to get a continuous extension fˆ: E Ñ RN . What we are
actually going to show now is that
pC pxq ,
fˆpxq R N
for every x P BC .
34
pC pxq, for every x P F . On the other
We already know by (N3) that fˆpxq R N
hand, if x P BCzFδ , it is sufficient to combine (N1) and (N2). Let us now
pC pxq. Then, since
take x P Fδ zF and assume by contradiction that fˆpxq P N
pC pxq is a convex cone and αpxq P N
pC pxq,
N
f pπF pxqq “
λx
1 ˆ
pC pxq ,
f pxq `
αpxq P N
1 ´ λx
1 ´ λx
pC pxq. The lemma is thus proved.
a contradiction with f pπF pxqq R N
We can now proceed to complete the proof of our theorem.
Let E “ D Y C1 Y ¨ ¨ ¨ Y CM be an optimal reconstruction of the truncated convex body D. Applying iteratively Lemma 24 to each single partial
reconstruction Ci , we obtain a continuous extension fˆ: E Ñ RN such that
fˆpxq R NCi pxq ,
for every x P BCi ,
for i “ 1, . . . , M , and hence also
fˆpxq R NE pxq ,
for every x P BE .
Thus, by Theorem 3, we have that
degpfˆ, Eq “ degpfˆ, C1 q “ ¨ ¨ ¨ “ degpfˆ, CM q “ p´1qN .
By the additivity property of the topological degree, we have
degpfˆ, Dq “ degpfˆ, Eq ´
M
ÿ
degpfˆ, Ci q “ p´1qN p1 ´ M q .
i“1
Since f coincides with fˆ on D, the theorem is proved.
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