characterization of almost convex functions
The proof is based on some simple observations about the values of an almost convex function. Suppose that and . Then for any , it must be the case that . This follows from the fact that, by definition of almost convex, either or . Since the first option is excluded by assumption, the second option must be true.
Furthermore, with , as above, is nondecreasing in the half-line . By the result of the last paragraph, it suffices to show that is non-decreasing in the open half-line . This is tantamount to showing that, if , then . From the conlusion of last paragraph, we already know that . Applying the result shown in the last paragraph to this conclusion, we further conclude that , as desired.
By replacing “” by “” in the above two paragraphs suitably, we also can likewise that, if and , then is nonincreasing on the half-line .
Now assume that is almost convex but not monotonic. By the hypothesis of nonmomotonicity, there must exist such that it is the case that neither nor . Furthermore, by almost-convexity, it follows that and . This, in turn, implies that is nonincreasing on and nondecreasing on .
Let be the set of all real numbers such that is nondecreasing on the interval . This set is not empty because . It is a proper subset of the real line because, for instance, whenever . This follows from the observation that cannot be nondecreasing on because . Also, must be a proper subset of the real line, because, if it were not, would be nondecreasing on the whole real line, which is contrary to assumption.
Note that, if and , then as well. This is an expression of the fact that, if a function is not monotonic on a set, it is not monotonic on a superset, which is the contrapositive of the assertion that a the resticition of a function which is monotonic on a set to a subset is still monotonic. Since there exists a real number such that , this means that is a lower bound for . Since is bounded from below and not empty, it follows that has a greatest lower bound, which we shall call .
By construction, is non-decreasing on the half-line . We will now show that is nonincreasing on the half-line . Suppose that . Then, by the choice of , the function is not nondecreasing on the half-line . This means that there must exist such that and . By the result demonstrated above, it follows that is nonincreasing on , hence, since , in particular, is nononicreasing on . Since is nonincreasing on for all , it is the case that is nonincreasing on .
|Title||characterization of almost convex functions|
|Date of creation||2013-03-22 15:21:07|
|Last modified on||2013-03-22 15:21:07|
|Last modified by||rspuzio (6075)|