The set X is convex if for any x, y from X, we have that the line segment joining x and y: ax + (1-a)y, also belongs to X, for any scalar a from (0, d], d > 0.
Erm, yeah, but it appears obvious. If a and b are in S, then the line segment between them is in X, hence the image of the line segment is a convex subset of R, a and b both satisfy f(a) and f(b) <=c so, I repeat, what does a convex subset of R look like?
EDIT think i have a different notion of a convex function than you. i'm guessing you mean that f is convex if for each a and b and x any point on the line segment a to b then f(x) < = (f(a)+f(b))/2, but that makes it even easier.
I think I am losing the point here, sorry about that. So, the point here is that, if S is a convex subset of R, and X is a convex subset of R, and for both of them exists a function f, then for any two points x and y from X, they also belong to S such that S = {x from X: f(x)<=c}.
In such a way that:
f(ax+(1-a)y)<=af(x)+(1-a)f(y)
then
<=ac+(1-a)c = c
for which f(x)<=c, this implies that S is convex.
Right? Sorry if I am wasting you time... =S
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