ON SOME CRITERIA OF CONVEXITY FOR COMPACT SETS

Abstract

We establish some criteria of convexity for compact sets in the Euclidean space. Analogs of these results
are extended to complex and hypercomplex cases.

Full text

Ukrainian Mathematical Journal, Vol.
63, No.
4, September, 2011 (Ukrainian Original Vol.
63, No.
4, April, 2011)
ON SOME CRITERIA OF CONVEXITY FOR COMPACT SETS
Yu. B. Zelins’kyi, I. Yu. Vyhovs’ka, and M. V. Tkachuk UDC 517.5+513.835
We establish some criteria of convexity for compact sets in the Euclidean space. Analogs of these re-
sults are extended to complex and hypercomplex cases.
In the present paper, we generalize the Aumann theorem [1] to the case of acyclic compacts sets and estab-
lish the convexity of one class of hypercomplex convex polyhedra in the many-dimensional hypercomplex (qua-
ternionic) space
H
n
. The terms used in the present paper without explanation are taken from the monographs
[2, 3].
Lemma 1. Let
K⊂
n
be an acyclic nonconvex compact set. Then there exists a supporting hyperplane
for K whose intersection with K is a set containing a nonzero cycle.
Proof. Assume that this is not true and the intersections of all supporting planes with K are acyclic sets.
Without loss of generality, we can assume that the interior of the convex hull of a compact set is nonempty (if
this is not true, then we can pass to the smallest plane containing this set) and the origin of coordinates lies in the
interior of the convex hull
conv K
of the compact set K.
Let
(conv K)*
be a polar to
conv K
, which is a compact set under the assumptions imposed on K. In
the space

n
×
n
, we consider the set
conv K×(conv K)*
or, more exactly, its subset
F
which can be de-
fined in two equivalent ways
F=(x,y)∈conv K×(conv K)* x∈∂ conv K,y∈∂ (conv K)*
[]
{}
,
where the point
y
defines a supporting hyperplane for
conv K
passing through the point x or, which is the
same, the point x defines a supporting hyperplane for
(conv K)*
passing through the point
y
. In view of the
convexity of the sets
conv K
and
(conv K)*
, supporting hyperplanes intersect these sets on convex and,
hence, acyclic sets. Thus, there exist two continuous acyclic mappings
p
1
:F→∂(conv K)
and
p
2
:F→
∂[(conv K)*]
(shrinking the corresponding intersections with supporting planes into points). By virtue of the Vietoris–Begle
theorem, they induce isomorphisms of the corresponding groups of cohomologies. Since the boundaries of con-
vex nondegenerate compact sets are homeomorphic to an
(n−1)
-sphere, the corresponding groups of the set
F
coincide with the groups of the sphere. Further, we use the equality of polars
K*=(conv K)*
. We now con-
sider a subset of the set
F
with the following properties:
Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, Ukraine.
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 4, pp. 466–471, April, 2011. Original article submitted December 30,
2010.
538 1072-3374/11/6304–0538 © 2011 Springer Science+Business Media, Inc.

ON SOME CRITERIA OF CONVEXITY FOR COMPACT SETS 539
F0=(x,y)∈Fx∈K∩∂conv K,y∈∂ (conv K)*
[]
{}
,
where, as above, the point
y
specifies the supporting hyperplane for K passing through the point x and the
point x specifies the supporting hyperplane for
(conv K)*
passing through the point
y
. The inequality
K≠
conv K
implies that
F
0
is a proper subset of
F
. As above, there exist continuous mappings
h
1
:F
0
→
K∩∂(conv K)
and
h
2
:F
0
→∂[(conv K)*]
.
It is necessary to show that they are acyclic. The projection
h
1
is acyclic because the set of supporting
planes at the point
x∈K
coincides with set of supporting planes at the same point for
conv K
. The projection
h
2
is acyclic because, according to the assumption, the intersections of the compact set K with supporting hy-
perplanes are acyclic. Thus, as above, we have two acyclic mappings inducing isomorphisms of the groups of
cohomologies. Therefore,
H
n−1
(K∩∂conv K)≈H
n−1
(F
0
)≈H
n−1
(conv K)*
[]
≈H
n−1
(S
n−1
)
.
This is possible only in the case where
K∩∂conv K=∂conv K
and, hence, in view of the acyclicity of
K, we conclude that
conv K=K
. The obtained contradiction proves the lemma.
Example 1. Consider the hemisphere
S
−
={(x
1
,x
2
,x
3
)x
1
2
+x
2
2
+x
3
2
=1,
x
3
≤0
}
.
The supporting plane
x
3
=0
intersects it over a one-dimensional cycle (circle). The intersections with the
other supporting planes are the corresponding unique points of the hemisphere.
Note that, in the proof of the lemma, we use the property of acyclicity of a compact set only at the end of
the proof (in showing its convexity). Thus, the following corollary is true:
Corollary 1. Let
K⊂
n
be a compact set. If each supporting hyperplane for K intersects K over an
acyclic set, then either K is a convex compact set or it contains a nonzero
(n−1)
-cycle.
If K has internal voids, then the support of this cycle is the boundary
∂conv K
.
Definition 1. An
m
-plane
L
,
0≤m≤n−1
, is called supporting for the compact set K if
L∩K⊂
∂K
.
Lemma 1 yields the following generalization of the Aumann theorem for acyclic compact sets:
Theorem 1. For an acyclic compact set
K⊂
n
to be convex, it is necessary and sufficient that all its in-
tersections with supporting
m
-planes,
1≤m≤n−1
, be acyclic.
This result can readily be generalized to the complex case. The terms used in what follows are defined
in [2].

540 YU. B. ZELINS’KYI, I. YU. VYHOVS’KA, AND M. V. TKACHUK
Proposition 1 [2, p. 132]. Compact sets
K⊂C
n
whose intersections with supporting hyperplanes are
connected have strongly linear convex
c
-hulls.
Theorem 2. For an acyclic compact set
K⊂C
n
with nonempty interior to be strongly linearly convex, it
is necessary and sufficient that all its intersections with complex supporting
m
- planes,
1≤m≤n−1
, be
acyclic.
The proof of the theorem repeats the reasoning from the proof of Lemma 1 and Theorem 1 with the use of
the set conjugate to K. The obtained result improves Theorem 9.1 in [2] in the case of acyclic compact sets.
In what follows, we generalize Theorem 11.1 in [2] to the hypercomplex case.
Definition 2. The set
E⊂H
n
is called linearly hypercomplex convex if its complement to
H
n
is a union
of hyperplanes from the space
H
n
.
In view of the noncommutativity of multiplication in the field of quaternions, the planes defined by multi-
plication from the left and from the right are different. In what follows, we consider linear subsets defined by
multiplication from the left (in the other case, the reasoning is similar).
Definition 3. A hypercomplex polyhedron is defined as a set of the form
E=z
{
:f
j
h
()
∈E
j
,j∈J=1, 2, …,N
}
{
}
,
where the sets
Ej⊂H
and
f
j
(h)=a
jk
h
k
k=1
n
∑
.
Moreover, any two functions
fj
and
f
k
,
k≠j
, are linearly independent and each function
fj
maps
E
into
a subset
E
j
of a hypercomplex straight line. The sets
E
j
are called generatrices of the hypercomplex poly-
hedron
E
.
It is easy to see that every hypercomplex polyhedron
E
is a linearly hypercomplex convex set. A point
h
0
=(h
1
,h
2
,…,h
n
)
does not belong to the hypercomplex polyhedron
E
if and only if there exist a natural
number
j∈J
and a quaternion
b
different from the divisors of zero such that
f
j
(h
0
)=a
jk
h
k
k=1
n
∑=b∉E
j
f
j
(h)
.
Then the hyperplane
ajk hk=b
k=1
n
∑
passes through the point
h
0
and, clearly, does not intersect
E
.
Definition 4. The set

ON SOME CRITERIA OF CONVEXITY FOR COMPACT SETS 541
Γ
j
=h∈∂E:f
j
(h)
{
∈∂E
j
and f
k
(h)∈intE
k
,k=1, 2, …,j−1, j+1, …,N
}
is called a face of the polyhedron
E
and the set
ˆ
E=h
{
:f
j
h
()
∈∂E
j
,j∈J=1, 2, …,N
}
{
}
is called the skeleton of the polyhedron
E
.
In what follows, we consider a special case of hypercomplex polyhedra in the form of nondegenerate Eu-
clidean products lying in hypercomplex one-dimensional spaces, i.e., none of the factors is a point or the entire
space.
Definition 5. The set
E⊂H
n
is called strongly linearly convex if its intersections by hypercomplex
straight lines are acyclic (homotopically equivalent to a point).
In [4], it is shown that strongly linearly convex domains and compact sets are linearly hypercomplex convex
sets.
The theorem presented below characterizes strongly linearly convex compact sets representable in the form
Cartesian products.
Theorem 3. A compact set in the form of nondegenerate Cartesian product is strongly linearly convex if
and only if it is convex.
Proof. Let
E
be a strongly linearly convex compact set. Without loss of generality, we can assume that
E=
E
1
×E
2
⊂H
2
. Otherwise, it suffices to pass to the intersection of
E
with a two-dimensional hypercom-
plex space.
To complete the proof of the theorem, we need several lemmas.
Lemma 2. Let
K
1
and
K
2
be two compact acyclic subsets of the Euclidean space

4
each of which
contains at least two points. Moreover, the set
K
2
is a convex body (i.e., the closure of an open set). If, for
any nondegenerate linear mapping
f
, the set
K
1
∩f(K
2
)
is acyclic (possibly empty), then the set
K
1
is also
convex.
Proof. Assume that the set
K
1
is nonempty. If the set
K
1
is not connected, then we choose two points
x
1
and
x
2
lying in different components of
K
1
. One can easily find a linear mapping
f
that maps two points
of the compact set
K
2
into the indicated two points of the set
K
1
. However, since
f
is homeomorphic, the
intersection
K
1
∩
f(K
2
)
is not connected, which contradicts the condition.
If the set
K
1
is connected, then, according to Lemma 1, there exists an intersection of
K
1
with a support-
ing hyperplane containing a nonzero cycle. A mapping
f
is chosen to guarantee that the interior of
f(K
2
)
contains a ball of sufficiently large radius and that a hyperplane parallel to the tangent in a sufficiently small vi-
cinity of the point of tangency contains, in the intersection with
f(K
2
)
, a ball whose diameter is greater than
the diameter of the intersection of
K
1
with the selected hyperplane. One can now readily choose a shift of the
set
f(K
2
)
such that the major part of this set and the set
K
1
are located in different half spaces, and the inter-

542 YU. B. ZELINS’KYI, I. YU. VYHOVS’KA, AND M. V. TKACHUK
section
K
1
∩f(K
2
)
insignificantly differs from the intersection of
K
1
with the chosen supporting plane and,
hence, contains a nonzero cycle. Thus, the convexity of the set
K
1
is established.
It is easy to see that the lemma remains valid if a four-dimensional Euclidean space is regarded as a hy-
percomplex space
H
and the mapping
f
is regarded as a hypercomplex linear mapping.
We now show that Lemma 2 is not true if we restrict ourselves to the case of linear shifts without ho-
motheties.
Example 2. Let
K
2
be a unit ball and let
K
1
be a hemisphere of unit radius. If we restrict ourselves to
the case of linear translations of the set
K
2
without homotheties, then all possible pair intersections
K
1
∩f(K
2
)
of the sets are acyclic.
Definition 6. A 1-flag in the space

n
is defined as a closed subspace of this space and its boundary is
called the pole of the flag. We extend this definition by induction: The union of an open subspace with the
(k−1)
-flag lying in its boundary is called a
k
-flag in the space

n
,
1≤k≤n
. Moreover, the
k
-flag contains
all smaller
m
-flags,
1≤m≤k
, and its pole is defined as the pole of the smallest embedded 1-flag. For the sake
of completeness, we say that the space

n
itself is both the 0-flag and the pole of this flag. Two flags are
called complementary if their intersection is contained in the pole.
Lemma 3. Every compact set
K⊂
n
lies in a certain n-flag and its intersection with each smaller sub-
flag is not empty.
The lemma is proved by subsequent exclusion from the space its parts lying on the other side of the corre-
sponding supporting planes.
Lemma 4. For each nonconvex compact set
K⊂
n
, there exists an imbedding into a certain
k
-flag
such that its intersection with each smaller subflag is nonempty and the intersection with the pole is the support
of an
(n−k−1)
-measurable cycle.
Proof. According to Corollary 1, if the intersection with the pole, which is an
(n−k)
-dimensional Euclid-
ean space, contains a cycle of dimensionality lower than
n−k−1
, then there exists an imbedding of the com-
pact set into a
(k+1)
-subflag, etc., until we get a nonzero cycle of the maximum possible dimensionality in the
pole.
Lemma 5. Let
K
1
,K
2
⊂
n
be two compact sets containing nonzero
(n−1)
-cycles and the origin of co-
ordinates,
n≥2
. Then there exists a mapping
f
such that
K
1
∩f(K
2
)
is the support of a nonzero cycle,
where
f
is the superposition of rotation of the space

n
and its homothety about the origin of coordinates.
Proof. In view of the conditions of the lemma and the Künneth formula [5] in the form of the exact se-
quence of the groups of cohomologies
0→H
n−1
(K
1
)⊗H
n−1
(K
2
)→H
2n−2
(K
1
×K
2
)
,
we conclude that the
(2n−2)
-dimensional group of cohomologies of the Cartesian product
K
1
×K
2
is

ON SOME CRITERIA OF CONVEXITY FOR COMPACT SETS 543
nonzero. The graph of the mapping described in the lemma is the intersection of this Cartesian product with an
n-
dimensional plane passing through the origin of coordinates. Assume that, for each mapping
f
, the sets
K
1
∩f(K
2
)
are acyclic. The union of the set of intersections coincides with the Cartesian product
K
1
×K
2
.
Further, by analogy with Theorem 4.2 in [3], we demonstrate that
K
1
×K
2
is acyclic, which contradicts the
facts established above.
We now return to the proof of Theorem 3. If one of the factors is convex, then, in the case where it contains
interior points, it suffices to apply Lemma 2 to obtain its nonacyclic intersection with a hyperplane. If it does
not contain interior points, then it is located in the real hyperplane of the space
H
. Hence, it can be placed in
the supporting hyperplane for
K
1
intersecting
K
1
over a cycle and, by using the homothety of
K
2
, it is possi-
ble to get a cycle in the intersection
K
1
∩f(K
2
)
equivalent to the intersection of the original compact set with
a hypercomplex straight line.
If both factors are not convex, then, according to Lemma 4, we place these factors in flags so that the cycles
of maximum dimensionality lie in the corresponding poles and the interiors of the compact sets lie in the com-
plementary flags. If the dimensions of these poles are different, then we fix the lower dimensionality (denoted
by
m
) and, in the pole of higher dimensionality, select a subspace of the indicated dimensionality split by the
cycle of the pole. This is possible because, as shown above, the nonzero cycle splits the pole. Hence, by Lem-
ma 5, it suffices to select a proper mapping in the Euclidean space

m
and the interiors of the compact sets
will be disjoint. Thus, the existence of nonacyclic intersections of the subsets of poles established in Lemma 5
contradicts the strong hypercomplex convexity of the compact set K, which completes the proof of the theo-
rem.
For the criterion of convexity of a domain in the Euclidean space, see [6].
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