characterization of almost convex functions

A real function f is almost convex iff it is monotonicPlanetmathPlanetmath or there exists p such that f is nonincreasing on the half-line (-,p) and nondecreasing on the half-line (p,+)


The proof is based on some simple observations about the values of an almost convex function. Suppose that a<b and f(a)f(b). Then for any c>b, it must be the case that f(b)f(c). This follows from the fact that, by definition of almost convex, either f(b)f(a) or f(b)f(c). Since the first option is excluded by assumptionPlanetmathPlanetmath, the second option must be true.

Furthermore, with a, b as above, f is nondecreasing in the half-line [b,). By the result of the last paragraph, it suffices to show that f is non-decreasing in the open half-line (c,). This is tantamount to showing that, if c<d<e, then f(d)f(e). From the conlusion of last paragraph, we already know that f(c)f(d). Applying the result shown in the last paragraph to this conclusionMathworldPlanetmath, we further conclude that f(d)f(e), as desired.

By replacing “” by “” in the above two paragraphs suitably, we also can likewise that, if a<b and f(a)f(b), then f is nonincreasing on the half-line (-,a].

Now assume that f is almost convex but not monotonic. By the hypothesis of nonmomotonicity, there must exist a<b<c such that it is the case that neither f(a)f(b)f(c) nor f(a)f(b)f(c). Furthermore, by almost-convexity, it follows that f(b)f(a) and f(b)f(c). This, in turn, implies that f is nonincreasing on (-,a] and nondecreasing on [c,+).

Let L be the set of all real numbers q such that f is nondecreasing on the interval (q,+). This set is not empty because cL. It is a proper subsetMathworldPlanetmathPlanetmath of the real line because, for instance, qL whenever q<a. This follows from the observation that f cannot be nondecreasing on (q,+) because f(a)>f(b). Also, L must be a proper subset of the real line, because, if it were not, f would be nondecreasing on the whole real line, which is contrary to assumption.

Note that, if r<q and qL, then rL as well. This is an expression of the fact that, if a functionMathworldPlanetmath is not monotonic on a set, it is not monotonic on a supersetMathworldPlanetmath, 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 r such that rL, this means that r is a lower bound for L. Since L is bounded from below and not empty, it follows that L has a greatest lower bound, which we shall call p.

By construction, f is non-decreasing on the half-line (p,+). We will now show that f is nonincreasing on the half-line (-,p). Suppose that q<p. Then, by the choice of p, the function f is not nondecreasing on the half-line (q,+). This means that there must exist a,b such that q<a<b and f(a)>f(b). By the result demonstrated above, it follows that f is nonincreasing on (-,a), hence, since q<a, in particular, f is nononicreasing on (-,q). Since f is nonincreasing on (-,q) for all q, it is the case that f is nonincreasing on (-,p).

Title characterization of almost convex functions
Canonical name CharacterizationOfAlmostConvexFunctions
Date of creation 2013-03-22 15:21:07
Last modified on 2013-03-22 15:21:07
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 9
Author rspuzio (6075)
Entry type Theorem
Classification msc 26A51