An extreme set of a subset 
of a vector space 
over 
or 
is a nonempty subset 
if no point of 
is an interior point of any line
segment whose endpoints are in 
except when both endpoints are
in 
. Said another way, 
is an extreme set of 
if whenever 
and
 |
for 
, it necessarily follows that
.
In the event that 
consists of a single point 
of 
,
is called an extreme
point of 
.
Extreme points play an important role in a number
of areas of math, e.g., in the Krein-Milman theorem
in functional analysis.
