if S is a set in the euclidean vector space U with inner product and norm defined in it, a point x in S is called an extreme pt of S if x is not interior to any segment contained in S.
is there an intuitive way of thinking about this?
1. what would be an example of bounded and convex set without extreme points?
2. suppose S is a polyhedron. is it true in general that if x in S is not interior to S, then x is an extreme pt of of S?
3. given linear functionals g_k:R^2->R, and real numbers b_k with k=1,2,3,4, and polyhedron S determined by g_k(x)<=b_k, is it possible for S to have 7 or more extreme pts? if not, how many extreme points can S have at most?
i have an idea for #1 (would it be ok to consider some open set in R^2? say an open halfspace defined by x>0) , but i don't really understand 2 and 3 (even though i have this feeling that 2 must be false). can someone help me?
the definition of polyhedron i have is that a polyhedron is an intersection of finitely many closed halfspaces (closed/convex set)
the way closed half space is defined is, let U be a euclidean vector space, and recall if g:U->R is a linear functional and if b is a real number, then the set {x:g(x)<=b} is called a close half space.
how would change of definitions change our answer?
thanks for help on 1 and 2, i was able to successfully prove those problems. but for part c, is the correct answer 4? i've been drawing a bunch of lines and it seems like at most 4 vertex points can be attained from the intersection of four closed half-spaces. is my logic wrong?