I
A PROPERTY OF NONSEPARATED CONVEX SETS1 HUBERT HALKIN
Introduction. The theory of convex sets, and more particularly their separation properties, is a very useful tool in the solution of many optimal control problems. In the case of linear problems the theory of convex sets gives an immediate answer, see for instance LaSalle [l] and Neustadt [2]. When the problem is nonlinear it is sometimes useful to study its linearization and to prove that some results obtained for this linearization are indeed valid for the nonlinear problem itself. In many instances this proof can be carried out by elementary topological arguments. Pontryagin et al, pp. 97 and 111, outline such a proof using the concept of topological index. In the present paper we prove a more general result of the same type using Brouwer's fixed point theorem. We shall prove that
Theorem I. Let Pi and K2 be two nonseparated convex sets in a Euclidean space En. We assume that 0EKiC\K2 and 0EKif\K2. Let L be a positive constant. For every n > 0 we are given a continuous mapping jfrom Ki into En and a continuous mapping 0 and for i=l, 2 we have
(1)
I 0
ior all i and j=l,
2, ■ ■ ■, n + l. Indeed for any pG(0, 1) let e,(p)
= c*+p(e,- —e*). For a small enough p we have gj(p) -e^p) >0 for all iand j = 1,2, • • • , ra+ 1 and none of the other properties are violated. Since \(AAC\ 0
is not empty
which a fortiori
implies
that
U (0
is not empty. Let Xi, x2, ■ ■ ■ , xn+i be vectors
(11)
Xi = ( e„-)
(12)
= (-a, \
by
for i = 1, 2, ■ ■ ■, k
l
)
for i = k + 1, • • • , n + 1.
(w + 1 — k)\i/
It is easy to prove that
independent
in E"+1 determined
the vectors
xi, x2, ■ ■ ■ , x„+i are linearly
and that
(13)
Y\iXi=(0,0,---,0,2). «=i
Let A be the subset of P"+1 defined by r n+l
n+l
\
A = < Y M.x.: S Mi< 1, Mi> 0 V t-l
for/ = 1, 2, • ■■, n + 1>
.'-1
'
and let x* = (0, 0, • • ■ , 0, 1). We have x*Gint A. For any t;>0
let "be a continuous
mapping
from A into En+1
defined by
(14)
+ (-*( \
E *#), E \ i=A+i
/
w
)•
0 the mapping 'is well defined since the representation Ya-t' UiXi is unique (the vectors Xi, x2, ■ ■ ■ , xn+i are linearly
independent)
and since E*-i utfiEAi
and E ?-t+iM 0 and a point y interior (21)
Theorem
to the set K such that
cbiy) = ax.
I*. Let Ki and K2 be two nonseparated
convex sets in an
Enclidean space P". We assume that 0EKif~\K2 and 0EKiC\K2.
Let L
be a positive constant. We are given a continuous mapping
