characterization of almost convex functions

characterization of almost convex functions

A

real function

f

is almost convex iff it is

monotonic

or there exists

p

∈

ℝ

such that

f

is

nonincreasing

on the half-line

(

-

∞

,

p

)

and
nondecreasing on the half-line

(

p

,

+

∞

)

Proof:

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

assumption

, 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

conclusion

, 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

c

∈

L

. It is a

proper subset

of the real line because, for

instance

,

q

∉

L

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

q

∉

L

, then

r

∉

L

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

r

such that

r

∉

L

, 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

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