A****s
发帖数: 129 | 1 statement: in an open interval (a,b), f(x/2+y/2)<=1/2f(x)+1/2f(y)
f (a,b) to R continous --> f is convex.
sketch of proof: consider the function of t, t in (0,1), p,q constants in(a,
b):
g(t)=f(tp+(1-t)q)-tf(p)+(1-t)f(q). g is continuous.
g^-1((0,inf)) is open since (0,inf) is open, i.e., the points in (0,1)
which make g>0 are indeed a family of disjoint open intervals.
suppose for some p,q,and t' such that g(t')>0. then t' is in one of the open
intervals. then for those two endpoints of this op |
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