Page 433 - Advanced Linear Algebra
P. 433
Positive Solutions to Linear Systems: Convexity and Separation 417
exercise to show that any hyperplane has the form >²5Á 5)) ³ for an
appropriate vector .
5
A hyperplane defines two closed half-spaces
> s b ²5Á ³ ~ ¸% º5Á %» ¹
> s c ²5Á ³ ~ ¸% º5Á %» ¹
and two disjoint open half-spaces
> k b s ²5Á ³ ~ ¸% º5Á %» ¹
> k c s ²5Á ³ ~ ¸% º5Á %» ¹
It is clear that
> > b ²5Á ³ ~ > c ²5Á ³ q ²5Á ³
and that the sets > k b , > ²5Á ³ k c and ²5Á ³ form a partition of s .
> ²5Á ³
If 5 s and ? s , we let
º5Á ?» ~ ¸º5Á %» % ?¹
and write
º5Á ?»
to denote the fact that º5Á %» for all % ? .
@
Definition Two subsets and of s are strictly separated by a hyperplane
?
@
> ? lies in one open half-space determined by >²5Á ³ if ²5Á ³ and lies in
the other open half-space; in symbols, one of the following holds:
1) º5Á?» º5Á@ »
2) º5Á@ » º5Á?»
Note that 1) holds for and if and only if 2) holds for c5 and c , and so we
5
need only consider one of the conditions to demonstrate that two sets and @
?
are not strictly separated. Specifically, if 1) fails for all 5 and , then the
condition
ºc5Á @ » c ºc5Á ?»
also fails for all and and so 2) also fails for all and , whence and @
?
5
5
are not strictly separated.
Definition Two subsets and of s are strongly separated by a hyperplane
?
@
>²5Á ³ if there is an for which one of the following holds:
1) º5Á ?» c b º5Á @ »
2) º5Á @ » c b º5Á ?»
