I ntern. J. Math. Math. Si.
Vol. 4 No. 3 (1981) 513-528
513
LNC POINTS FOR m-CONVEX SETS
MARILYN BREEN
Department of Mathematics
The University of Oklahoma
Norman, Oklahoma
U.S.A.
73019
(Received April 8, 1980 and in revised form September 4, 1980)
ABSTRACT
S
Let
dimensional, with
Q
U
k
a finite union of
For
N
q
bdry S
conv(Q
q E Q
compact, if to every
N) c S
of
q
is an
u
Q
q
k
k > 0
there corresponds a
then
Q
may be written as
(d- 2)-manifolds.
S
any inc point of
Q
If
(d- 2)-dimenslonal manifolds, each
or fewer
is an essential inc point of order
q
such that
S
d-
locally a full
then for some neighborhood
is expressible as a union of
For
S
S
k
q
d
the corresponding set of Inc points of
essential inc point of order
containing
R
m-convex subset of
be closed,
N
a convex neighborhood of
q
Q
is nowhere dense in bdry
S
Moreover, if
Q 0 N
is not homeomorphic to a
(d
l)-dimensional
Points
of nonconvexity,
That is,
then
and
manifold.
KEY WORDS AND PHRASES.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
m-Convex Ss.
Primary 52.A20, 52.A40.
INTRODUCTION.
Let
S
be a subset of
if and only if for every
m
d
R
S
The set
is said to be
distinct points in
line segments determined by these points lies in
is not
j-convex for
j < m
then
S
is
S
S
m
at least one of the
S
If the
exactly m-convex
is said to be a point of local convexity of
m-convex,
>-
()
m-convex set
A point
x
2
in
if and only if there is some
S
S
514
M. BREEN
N
neighborhood
of
such that if
x
fails to be locally convex at some point
of local nonconvexity (inc point) of
N
S
y,z
q
S
in
[y,z]
then
then
c S
S
If
is called a
q
point
S
Few studies have been made concerning points of local nonconvexity for
Valentine [3] has proved that for
convex sets.
of
d
R
Q c int conv S
and
(d- 2)-dimensional manifolds.
m-convex sets.
For points
The following familiar terminology will be used:
_x sees
in
S
Points
if for
i
aff S
S_
via
x.
n
i < j
conv S
S
_S
are visually independent via
does not see
int S
ker S
x,y
if and only if the corresponding segment
in
Xl,...,Xn
int ker S
if
The purpose of this paper is to obtain an
analogue of Valentine’s result for
we say
S
consists of a finite number of disjoint closed
Q
then
3-convex subset
a compact
the corresponding set of inc points of
Q
with
S
m-
x.
S
via
rel int S
in
S
Ix,y]
lies
if and only
Throughout the paper,
bdry S
and
cl S
will be
used to denote the affine hull, convex hull, kernel, interior, relative interior,
S
boundary, and closure, respectively, of the set
Also, for points
through
and for point
y
U{R(x,t)
S
2.
and set
will denote the ray emanating from
cone (x,T)
T
will be a closed subset of
dimensional- i.e., for
N)
x
R(x,y)
y
x
will represent
T}
t
Finally,
dim(S
and
x
d
And
d
which is locally a full
any neighborhood of
s
will denote the set of inc points of
S
s
Q
S
R
in
and
N
d-
K
ESSENTIAL LNC POINTS OF ORDER
We begin with the follDwing definitions for the closed set
Q
responding collection of inc points
S
and its cor-
The first definition is an adaptation
of Definition i in [i]
DEFINITION i.
q 6 Q
Let
there is some neighborhood
of
q
with
N c N’
DEFINITION 2.
(S
We say that
N’
of
N)
Q
We say that
q
q
s
essential if and only if
such that for every convex neighborhood
q
is connected.
Q
has order
k
if and only if there is
N
LNC POINTS FOR m-CONVEX SETS
N’
some neighborhood
of
such that the following are true.
q
N’) c S
i)
Conv(Q
2)
For every convex neighborhood
conv(Q
N)
N
of
S
N c N’
with
q
N)
(S
k-tuple of points which are visually
contains at least one
independent via
(k + l)-tuple of points visually independent
and no
S
via
3)
515
N
For every convex neighborhood
N’)
dim conv(Q
of
q
d
If this dimension is
q E int conv(C U (Q
N))
If this dimension is
d
N c N’
with
C
conv(Q O N)
(S 0 N)
of
q E Tel int (S
then
N)
then
for each component
i
conv(Q
dim
N))
aff(Q
The following lemmas will be useful.
LEMMA i.
Let
S
be a closed
ing set of inc points of
PROOF.
S
Q c cl(S
Then
N
of
Q)
(S
N
q
x
I
and
x
x
S
Q
M’
M
Q
in
M,M’
and let
S
q
N c Q
S
respectively so that no point of
I
the correspond-
Q)
Then
which are visually independent via
Q
with
Suppose on the contrary that for some point
neighborhood
of
d
R
m-convex set in
and some
Select
c N
be neighborhoods
Since
S
By an obvious induction, we obtain m visually independent points
x
via
S
Let
N
S
N
in
2
which are visually independent via
Our assumption is false and
S
m-convexity of
Let
Qx
y6 conv(Q
i _< i <_ k
denote the subset of
U {x}
x
x
U {x})
to prove that
y E conv{z I,
k _< d
finished, so assume that
above must be
Qx
N) c S
conv(Q
conv(Q
N)
which
x
sees
i
member
conv(Q x U {x}) c S
Carathodory’s theorem,
for
x
be a convex neighborhood for which
and let
Then
PROOF.
subset of
2
via
Q)
LEMMA 2.
let
x
contradicting the
Xl,X2,...,xm
Q c cl(S
choose
M’
sees any point of
S
I
in
Xl,X 1
y
If
...,Zk+ I}
y
and we may assume that
Further, we assume that
N)
y E
k
S
Then by
for an appropriate
cl conv(Q
cl conv(Q
y
N) c S
is minimal.
+
the argument is
Hence one of the
conv{x,z I
k
z
k}
Then
z
i
points
where
zl
Qx
M. BREEN
516
P
=_
conv{x, zl,
z
k}
is a
k-simplex having
y
in its relative interior.
Clearly the result is
We use an inductive argument to finish the proof.
I
k
true for
For
numbers less than
P
face of
y
S
cl conv(Q N N)
y
cl conv(Q
from
N)
H’
b
{z
{x,y} and
and clearly
strictly separating
Zk}
I
L
Then
P
L
H
[a,b]
is an interval
y
lie in facets of
P
Hence by our induction hypothesis,
{x}
lie on opposite sides of
Q N N
S
in
conv{x,a,b}
c S
point of
and
y
where
be
and
a
Ix,a] U Ix,b]
c S
so there can be no Inc
Therefore, by a lemma of Valentine [4, Corollary i],
conv{x,a,b}
Thus
H’
L
and let
y
and containing
with
Clearly
lie in opposite
H
be a hyperplane parallel to
H’
a line in
H
there is a hyperplane
open halfspaces determined by
Let
Thus we may assume that every proper
k
to prove for
k
lies in
Since
assume that the result is true for all natural
2
k
S
and the lemma is proved.
The following theorem is an analogue of Valentine’s result for
3-convex
sets.
Let
THEOREM i.
S
Q
d-dimensional, with
be a closed
Q
and 2.
(d
or fewer
of
q
is
q
2)-dimensional manifolds,
N’
be a convex neighborhood of
q
satisfying Definitions 1
N’)
CASE i.
Assume that for every neighborhood
dim conv(Q 0 M)
is
3-convex.
Q’
U
then for some neighborhood
k
If
The proof will require three cases, each determined by the dimension of
conv(Q
q
S
q
Let
PROOF.
of
k
is expressible as a union of
each containing
locally a full
S
the corresponding set of inc points for
an essential inc point of order
U
d
R
m-convex set in
cl(Q
2
Letting
N’)
of
q
S
Q’
N’
3-convex, and
is
S’
S’
is compact and
clearly
It is easy to show that every inc point of a
in the kernel of that set, so
Q’
c ker
S’
and hence
int ker
If the order
q
N’)
cl(S
denote the set of inc points of
M c N’
with
We proceed by induction on the order of
d
then
M
3-convex set lies
S’
*
Also,
LNC POINTS FOR m-CONVEX SETS
satisfies Definition 2,
q
since
U
there is a neighborhood
of
[3, Lemma 4
Thus by
u
Q
such that
q
S’
int conv
q
517
5],
and
(d-2)-dimensional
is a
manifold.
Inductively, assume that the result is true for order
order
k
q
V
that
q
(S
of
C
component
R(x,q)
so that
N
hood
conv({p} U V)
int
of
[x,r)
N c W
conv(Q
Let
We assert that
certainly
N
S
R(x,r)
W)
W)
and
A
supporting
conv(Q
W)
seen by
By the proof of
x
conv(T U {x})
and
r
T
r
so for some neighborhood
W)
r
r
at
A
of
conv(Q
then
S
A
A
H
A’
of
r
r
int
conv(A’
via
S
via
I
S
A’
conv(Q
int
A)
w)
x
with
select
and select
(v,w) c int(S
{x})
and since
c A
c int
W)
would
r
would be
w
A)
in
Let
v
in
(x,r)
H
in the open halfspace
A
r
But since
A)
int(S
A’
N
conv(T U {x}) c int K
x
A
S
be a hyperplane
H
I
determined by
Using Valentine’s lemma [4, Corollary i], it is not hard to show that
S
N
Qk
in
N
r
clearly
Without loss of generality, assume that
intersects
i convex,
r
Q
were not in
r
T
Q
in
is a point of
r
so
Now if
Q
K
the corresponding set of inc points of
Qk
W) 0 R(x,r) c int(S
conv(Q
S
sees
w)
denote the closure of the set
W)
S
R(x,r)
Then since
H
N c conv(Q
To obtain the reverse inclusion, for
Inc point of
Since
int
conv(Q
w)
bdry conv(Q
N
in
By our construction, for
N
convex and hence disjoint from
A
r
conv(Q
K
with
Qk
conv(Q
sees via
not be an
(s
x
for every
Finally, select a convex neighbor-
[x,r)]
N
Let
w)
T
Qk
r
x
which
Q
r
so
Qk
w)
W))
so
W)
conv(T U {x}) U conv(Q
r
conv(C U (Q
int
q
N’
p E
and select a point
we may select
[R(x,r)
W)
conv(T U {x}) c S
Lemma 2,
w)
conv(Q
int
denote the subset of
T
q
so that for all
conv(Q
int
[2, Lemma 2],
is starshaped relative to
N’)
Since
conv(Q
intersects
q
intersects
-_- W
w)
N’
S
int conv(Q
be a neighborhood in
to prove for
m-convex set is locally starshaped
Since a closed
without loss of generality assume that
Let
q < k
x
sees some neighborhood
this implies that
contradicting the fact that
M. BREEN
518
r E
Qk
We conclude that
Qk
N N
c Q
N T
the sets are equal, and our assertion
is proved.
To complete Case i, unfortunately it is necessary to examine two subcases:
the set
K
conv(T U {x})
If
CASE la.
U’
and let
S’
S’
M
so that
(S’
W)
conv(Q
y
cl(S
conv(Q
of
S’
w)
S’
C
If
And if
k
conv(Q
k- i
S’
then
Q’
Letting
k- i
T N U)
in
S
then
U
U) U
(Q’
(Qk
T
Q
of
Q’
q
U
2)-manifolds which contain
U’
N
U
N U)
q
2)-
(d
or fewer
k
is a union of
is
u
Q
Then
N
Qk
(d
or fewer
For simplicity of notation assume that
u) U (Q
W
denote the set of Inc points of
By an inductive argument, for an appropriate neighborhood
(Q’
_c
is an essential inc point
q
M which do not lie in
S
contains all inc points of
expressible as a union of
y
conv(Q q W)
[y,x]
N
Then for
[y,x]
If
c_
M
q
W)
M
S
of
visually independent points not in
i
is an inc point of
q
M
which contains
again impossible.
at most
of order at most
Q’
[y,x]
Hence
2)-dlmensional manifold.
conv(Q N w)
W)
cone(x,T)
impossible.
cone(x,T)
S’ N M has
(d
is a
(S
we assert that
y
_
[3, Lemma 4].
Select a convex neighborhood
C)
W)
then
would lie in
Thus
N U’
Qk
q
contains no point of
conv(Q
M)
of
then by a previous argument
d
satisfy the hypotheses of
denote the component of
C
Now let
x
q E K
and the point
for some neighborhood
has dimension
manifolds, the desired result.
If
S’
that
U
is convex,
_c
U’
N
u
Q
Then
Suppose that Case la does not occur.
CASE lb.
<_ d- i
dimension
N
U
select the neighborhood
U
Qk
U
is a
of
q
so
2)-
(d
This finishes ,Case la.
manifold.
Qk
S’
is not an inc point of
q
Q
T
it is clear that
conv(Q N w)
By a previous argument for some neighborhood
Also, since
Qk
N
dim
N
conv(Q
W)
d
dim conv(T
and
N
of
{x})
has
q
_< d
i
is exactly the set of points of intersection of
(conv(T U {x}))
with
Recall that
conv(T U {x})
Hence
N
is a neighborhood of
T
so
q
Qk
N
_c
Q
satisfying the definition of essential,
519
LNC POINTS FOR m-CONVEX SETS
so
(S
v,w
Select points
w E int
K
in
conv(Q N W)
W
v
tl,...,tn
[x,t I] U
[tl,t 2]
S
conv(Q q w)
r
conv(Q
W)
[x,r)
R(x,r)
Since
[x,t2] c
[x, tj] c_ S
may be used to show that
[x,tj_l _c
S
Q
t.
since
Q
j
dim conv(T
X
W)
and
U {x})
d
(tl,t 2)
N
we
Otherwise,
N c
(tl,t 2)
then
Then by a generalization of
t.
_c
[x,t 2]
S
j > 2
For
the above argument
An easy induction gives
Q
Thus
T c Q
and we know that
Let
Clearly
[x,r)]
[R(x,r)
so
intersects
S
W)
conv{x,t l,t 2
int
in
to
and assume that they
conv(Q
in
contradicting our choice of
Valentine’s lemma [4, Corollary i],
v
Furthermore, by our choice of
Q
bdry conv(Q
N
would lie in
r
from
Now by our definition
w
to
is the first point of
t.
assert that there can be no inc point
clearly
x
Q
into two disjoint connected sets.
N
separates
from
denote the consecutive vertices of
w
Then
S Q
and
(S N N)
X be a polygonal path in
w)
are labeled so that
j > i
_
v E (x,q)
with
v < q < w
is a path in
bdry conv(Q
of
N
Let
X U Ix,v]
Then
w
is locally convex and connected and hence polygonally connected.
N) Q
However, this is impossible
T
t.j
We conclude that Case Ib cannot occur,
and the previous argument in Case la guarantees our
result.
neighborhood of
Also assume that
Select
k
locally a full
(S
M)
H
M’
may be selected so that for
N’
c
and let
q
H)
rel int(S
M’
and
q
a neighborhood of
q
N’
Assume that
CASE 2.
M’)
dim conv(Q
H
aff(Q
M)
d
any convex
S
is starshaped relative to
visually independent points
x
q
,xk
I
(d
H c S
M
so without loss of generality we may assume that
M
be such
By Definition 2, we have
S
in
M
Since
S
is
d-dimensional, clearly these points may be selected in
For each
i
T.
consider the set
in
M
H
seen by
arguments used in the proof of Lemma 2, it is easy to show that
c S
M
Let
i
Also, using the definition of essential, one may show that
x
i
conv({xi}
T
i
By
T i)
is a
l)-dimensional set.
For simplicity of notation, assume that
q
is the origin in
R
d
and that
H
520
M. BREEN
is orthogonal to the vector
H
halfspaces determined by
S
i
e
labeled so that
z
where
-e
Q
i
6 H
i
and
I
U ((M
Ti)
e
z
S
of
for each
i
q
so that
each containing
CASE 3.
M
hood
x
Q D M
(d
is a
Qi
Ui
U
of
D Qi
Q
cl(Q Q M)
c
the subset of
S’
Let
Q Q M
S’
s’
Q’
S’
Q’)
Q’
r
S’
and
M
aff
Q’
A
Hence
Finally, select
A
x,y
(Clearly this is possible
is a
is a convex neighborhood of
S’
cl(int S’)
S’
aff
conv(Q D M)
which
S’
Q’
aff
S’
in
since
S’
conv{x,y,r}
there can be no inc point
p
c cl(S’
Q’
r
sees
r 6 Q
S’
[x,y]
cl(int S’).)
conv{x,y,r}
closed,
Also,
A
Let
denote
By repeating arguments in
S
with
[x,y]
S’
Q’)
is connected.
S’
is open and closed in
and
aff
must be some inc point in
in
A
satisfies
by [i, Lemma i].
to show that
sees via
r
Q’
M
Since
Q’
q
S Q M
We have
is connected.
S’
connected, so
be a point in
S’
Q D M
and hence
By a previous lemma,
is connected
c cl(S’
for some neighbor-
denote the closure of the set
[I, Lemma 5], it is easy to show that
that
2)-manfolds,
by a result in [i].
2
d
by the argument in [i, Lemma 4], the set
Now let
set.
Thus for an
(d
k
2
_< d
has dimension
the corresponding set of inc points of
connected, and
S.
lies
i
and it is easy to see that
is a union of
conv(Q D M)
we assert that
q
Definition i,
S
of inc points of
Qi
2)-dimensional manifold
U
q
conv(Q Q M)
satisfying Definition i.
Q’
2
is an inc point for some
Without loss of generality, assume that
S’
6 H
i
q
In case
of
if
I
Hence by Valentine’s theorem there is a neighborhood
i
convex set of dimension
so
Finally, define
I
[q,z])
H)
q 6 (int conv S i)
3-convex,
is
i
appropriate neighborhood
Q’
H
is in
I
it is easy to show that the set
i
Furthermore, every point of
int ker S
U
x
if
I
For each
Now
e
denote distinct open
to be the closure of the set
conv({x i} U
in
HI,H 2
Let
(i,0,...,0)
I
aff
S
Q’
and
aff
and
S
via
y
Q’
aff(Q’ U {x})
By Valentine’s lemma [4], there
but by our choice of
([x,y] U {r})
x
and
Otherwise,
y
521
LNC POINTS FOR m-CONVEX SETS
y E aff{p,x,r} c aff(Q’ U {x})
M) c Q
conv(Q
M
Hence
impossible.
must belong to
r
conv(Q
The reverse inclusion is obvious,
Q
and
Q N M
M)
and the assertion is proved.
S’
The set
is a closed connected set whose corresponding set of inc points
is convex and satisfies Definition I in [i].
in
[i],
Q’
2
d
has dimension
Hence by the corollary to Theorem 2
This completes Case 3 and finishes the proof
of the theorem.
COROLLARY i.
full
S
Let
be a compact
Q
d-dimenisonal, with
that for every point
q
in
tial inc point of order
k
R
m-convex set in
d
S
locally a
Assume
the corresponding set of inc points of
S
Q
is an essen-
there is some
Q
Then
k > 0
such that
q
(d
is a finite union of
2)-dimensional
manifolds.
PROOF.
Q
Since
is compact, the result is an immediate consequence of
Theorem i.
The following examples reveal that Theorem i fails in case
does not
q
satisfy both Definition i and Definition 2, part 3.
EXAMPLE i.
in Theorem i.
It is easy to find examples which show that
d >_ 3
For
meet in a single point
EXAMPLE 2.
dim conv(Q
S
I
simply consider two
d-dimensional convex sets which
q
To see that Definition 2, part 3 is required when
N)
d
2
d
let
and identify
R
2
with the complex plane.
be the infinite sided polygon having consecutive vertices
i
exp-
exp
(2 n-
Te
Clearly
Q N N
C
is
n >_ 0
set
of
S
S
Similarly, let
i
exp-,
exp 0
points are essential.
every component
l)i
2n
polygon with vertices
(See Figure 1.)
must be essential
q
cl(conv S
exp
1
5i
exp
8
U conv S 2)
conv(Q
N)
q
int
not expressible as a finite union of
example may be generalized to higher dimensions.
exp 0
be the infinite sided
(2 n+l
3)i
1
n
2n+l
3-convex, and its lnc
is
However, for every neighborhood
(S 0 N)
2
Let
N
of
q
exp i
conv(C U (Q
(d
and
N))
2)-manifolds.
The
M. BREEN
522
S
q
exp hi"
exp 0
Figure i.
To see that Definition 2, part 3 must be satisfied when
EXAMPLE 3.
conv(Q
dim
N)
In this plane let
complex plane.
vertices
v
separate
v
n
edge of
(2
x- y
H
plane
with the
3
and identify the
P
be the infinite sided polygon having
n
l)i
n >_ 1
At each vertex v
n >_ 0
n
2n
from the remaining vertices with a line L n so that L n
exp
n
P
d
let
d- i
(See Figure 2.)
L
and so that no two
v
n
adjacent to
Ln
Each line
strictly
cuts each
conv P
lines intersect in
n
Tn
determines a closed triangular subset
of
conv P
Let
R
be the rectangle in the
(i,-i)
(-i,-i)
(-i,0)
A
A
n
0
mod 3
or
n --_ i
2
U{Tn
n -_- 0
rood 3
or
n
S
cl A
I
=_
[e,e 3]
I
Clearly both
B
halfspaces determined by
(S
B
I
B
denote a
U S 2)
R
is interior to
S
I
U S2 U B
hood
N
of
U A0 U R
2 mod 3} U A
0
U{Tn
(0,0,0)
Let
n >_ i}
I
Finally, let
and
and define
U{Tn
A 0 =conv P
S
and
S
I
so that
I
cl A
2
S
and
S
=_
c_
mod 3}
[e,-e 3]
2
The set
B
conv(S I U S 2 U B)
cl H
and
I
S
2
_c
cl H
cl H
Hence letting
q
int
S
2
with
denote the
conv(S 0 N)
q
of inc points of
e
S
is exactly
(0,0,i)
3
Label the
2
may be constructed so that the point
it is not hard to show that
Q
where
are convex and closed.
2
3-dimensional parallelepiped in
Note that the set
(i,0)
plane whose vertices are
x- y
B
H
q
(-I,0,0)
4-convex set
for every neighbor-
523
LNC POINTS FOR m-CONVEX SETS
U {L i
N
of
T
q
mod 3}
0
i
i
dim
conv(Q
Definition 2 and
N)
N
Q
U [q,r]
I
d
yet
(i,0,0)
r
where
S
For every ne_onbehood
tisfy part B of
does not
(d
is not a finite union of
2)-manifolds.
N
more, it is interesting to notice that for every neighborhood
C
every component
conv(Q 0 N)
and
q
ment that
q
int
belong to
result in case
(-i,0)
(S
of
conv(Q
N)
conv(C U (Q
int conv(C
dim conv(Q
N)
N)
N))
U (Q
d
C
int
N))
conv(S
of
S
is exactl
urther-
N)
and for
N)
Thus the require-
io ot sufficient to guarantee our
i
q
(i,0)
(-i,-i)
r
(i,-i)
Figure 2.
The author would like to thank the referee for providing three additional
The first of these (Example 4) reveals that the conclusion
examples given below.
of Theorem 1 -y hold without Definition
EXAMPLE 4.
Let
S
The inc point
and parts 2 and 3 of Definitions 2 for
However,
Q
part 1
(S
be the closed set in Figure 3.
a smaller cube has been removed.)
hold.
J2
k
3"
q
S
of
is a cube from which
satisfies Definition i
Definition 2, part i does not
is expressible as a union of three
d
2
i
dimensional
524
M. BREEN
manifolds.
Whether Theorem I is true without Definition 2, part i remains an open
question.
Figure 3.
Furthermore, the conclusion of Theorem i can hold when
(Compare to Example i in which
as Example 5 reveals.
q
q
is not essential,
is not essential and
Theorem I fails.)
EXAMPLE 5.
For
d
2
let
S
be a union of two
(d- 2)-dimensional face
sect in a common
not essential, yet
Q
is a
Q
d-polytopes which inter-
Then the inc points of
S
are
(d- 2)-dimensional manifold.
It would be interesting to obtain an extension of Theorem I to include the
situations of Examples 4 and 5.
The final example by the referee illustrates Theorem i.
Let
S
be ’the union of four stacked cubes of equal size in
The point
q
is an essential inc point of order
EXAMPLE 6.
Figure 4.
expressible as a union of three
3
and
Q
1-dimensional manifolds, each containing
is
q
LNC POINTS FOR m-CONVEX SETS
525
Figure 4.
3.
O__ I__S NOWHERE DENSE IN BDRY
S
The final theorem will require the following easy lemma.
LEMMA 3.
Let
S
be a closed
exactly
PROOF.
Select
Nk_ I _c
I
N
x I,
By Lemma i, each N i
Xl,...,Xk_ I
i
N
S
If
(S
N) Q
is
which
N
S
be corresponding neighborhoods of
N
the corresponding
in
Xk_ I
yl,...,yk_ I visually independent
such that no point of
Q
be a convex neighborhood.
k-convex, then there exist points
are visually independent via
N
N
Let
S
set of inc points of
d
R
m-convex set in
N.
sees any point of
x
contains some point
3
i
Yl
N
S
via
and let
respectively,
Yk-i
via
S
i _< i < j _< k
S
Q
and the points
in
i
are the required visually independent points.
THEOREM 2.
S
Let
d-dimensional, with
Q
convex neighborhood of
be a closed
m-convex set in
the set of inc points of
N
q
bdry S
d
For
S
q
Q
Q
That is,
N
bdry S c Q
locally a full
S
R
in
Q
and
N
any
is nowhere dense in
bdry S
PROOF.
borhood
N
neighborhood
r
exists.
Assume on the contrary that
of
We assert that for some point
q
U
of
r
conv(Q
Select two points
x,y
U) c S
in
S ,q N
r
in
for some convex neigh-
Q
N
and some
Suppose on the contrary that no such
whose corresponding segment
[x,y]
526
S
is not in
[x,y]
The segment
x’
clearly we may select points
13 N
_c
2
bdry S
[x,y] with
[x’ ,y ]
x
x’
and
y
y’
[x,y]
and
N
N
N
x,y
N
so that no point of
N
bdry S
_c
conv(Q
N
Q
I
S
the fact that
r
N
Q
in
for
2
and
x
N l)
S
N
1
y
N2
and
conv(q
and
N
N 2)
is closed,
For
S
S
there
respectively,
S
via
Since
S
By an obvious induction,
2
S
visually independent points of
m
m-convex.
is
Since
sees any point of
1
Now repeat the argument for each of
we obtain a collection of
S
and since
in
exist disjoint convex neighborhoods
1
bdry S
intersects
,y
convenience of notation, assume
N
_ _
M. BREEN
contradicting
Hence our supposition is false and for some point
U
and for some neighborhood
of
U) c S
conv(Q
r
the desired
result.
Also assume that
x
exist points
N
S
and clearly at most one
conv(Q
point of
N)
via
of
2
M
S
conv(Q
x
0
1
i S j
sees any
N)
x
N
Yk
is in
x
conv(Q
N)
N)
2 _< i _< j
x
M
Q .)
z
1 <_ i <_ k
Yi
u)
conv(Q
N)
S
j
Let
N)
Qi
M c N
x
i
no point
by
xI
Replacing
N)
bdry S
sees
Hence in either case we have a collection of points
S
Qi
conv(Q
such that every point of
2
k
j
conv({Yi } U Qi _c
certainly dim conv({Yi }
would capture some point of
S
U c N
N
Q
for each
U Qi
Q
N)
bdry S
1
denote the subset of
By Lemma 2,
conv(Q
conv({y i} U
d
does not see any
conv(Q
For the moment, suppose that for every neighborhood
conv(Q
1
visually independent points,
j-convex, every point of
sees one of these points via
dim
x
delete
x 0 E (S N M)
Select
j
S
z
of
S
Now if every
via
bdry S
i
exists since
0
N)
xj_ 1
2
a collection of
xj_ 1
is exactly
(S
in
I
z E conv(Q
S
via
i
one of these points via
Yl
which are visually independent via
and for some neighborhood
x0,x 2
S
x
By Lemma 3, there
Q
sees one of
(Clearly such an
we have
and since
bdry S
Otherwise, some
from our listing.
S
point, say
x
3 _< j <_ m
j-convex,
(S N N)
in
xj_1
I
exactly
is
N) c S
conv(Q
Therefore, without loss of generality we may assume that
_< d
i
1
with
seen by
U
Q #
Yi
Since
for otherwise
in its interior, impossible
_
527
LNC POINTS FOR m-CONVEX SETS
N
Q
Thus
U c N
Moreover, since for every neighborhood
bdry S
sarily nowhere dense as a subset of
conv(Q
dim
S
Since
N
d-dimensional,
z E bdry S
to
For
is convex:
dim
conv(Q
u)
and for each
x
dim conv(Q
N)
and
and
Qi _c
U
dim(Q
dim
conv(Q
N)
U
i
1
d
Qi
some
N)
V
of
N
Q
cone(y I,QI ))
(S
conv({Yl}
d
having dimension at most
2
R
in
d
S
A parallel
bdry S)
d
and
must belong to
z
is indeed convex.
seen by
set, say
i _< k
i
Yi
QI’
each
impossible, our assumption is false, and
N
of
q
Since
zi
U{Ti
Q1
N
bdry S
1
Our previous argument
n}
i
Q
Zl,...,z n
Ti
seeing a subset
1
N
Q
d
has dimension
d-dimensional.
Q1
S
converging
is necessarily a convex subset of
Qi
with
then
interior to
may be repeated to obtain a finite set of visually independent points
in
i
i
bdry S
N
Q
d
d
z
{x }
n
dim conv(N
and
is a full
Q1
N)
We have a contradiction, and
N
Q #
U
we will show that
(Xn, p)
V
in
N c Q
for every
conv({Yl }
Then the set
p
denote the subset of
S
N
Q
for such a neighbor-
i
x < z < y
This implies that
(bdry S)
Qi
is a finite
Qi
dim conv(Q
there is a sequence
impossible.
z
Again let
and since
N
Q
in
and each
y
d
Hence
conv({Yi}
x,y
Xn
statement holds for
bdry S
bdry S c Q
N
x E bdry S
Since
is neces-
Qi
with
d
Otherwise, there would be a neighborhood
V c N
with
U c N
for some neighborhood
i
We assert that since
Q
does not
and by standard arguments
For convenience of notation, assume that
U
hood
U) _< d
is a full
i
We have a contradiction, our supposition is false,
bdry S
is nowhere dense in
U
N
Q
Q
U
That is,
Qi
Then
bdry S
union of sets, each nowhere dense in
Q
U
with
bdry S
U
lie in a hyperplane, it follows that
and
_< d
lies in a finite union of flats, each having dimension
of
Q1
Clearly this is
for every neighborhood
This completes the proof of Theorem 2.
Techniques identical to those employed in the proof of Theorem 2 may be used
to obtain the following result.
COROLLARY i.
Inc points of
S
Let
S
Then if
be a closed
conv(Q
m-convex set in
N) c S
R
d
with
for some neighborhood
the set of
Q
N
Q
N
M. BREEN
528
cannot be homeomorphic to a
(d- l)-dimensional manifold.
REFERENCES
i.
BREEN, Marilyn.
Points of Local Nonconvexity and Finite Unions of Convex
Sets, Canad. J. Math. 27 (1975), pp. 376-383.
Pm’
2.
KAY, David C. & GUAY, Merle D. Convexity and
J. Math. 8 (1970), pp. 39-52.
3.
VALENTINE, F. A.
4.
VALENTINE, F. A. Local Convexity and Ln Sets, Proc. Amer. Math. Soc. 16
(1965), pp. 1305-1310.
a Certain
Property
Israel
The Intersection of Two Convex Surfaces and Property
Proc. Amer. Math. Soc. 9 (1958), pp. 47-54.
P3’