Cone Monotonicity: Structure Theorem, Properties, and Comparisons to Other Notions of Monotonicity

In search of a meaningful 2-dimensional analog to mono- tonicity, we introduce two new definitions and give examples of and dis- cuss the relationship between these definitions and others that we found in the literature. Note: After we published the article in Abstract and Applied Analysis and after we searched multiple times for previous work, we discovered that Clarke at al. had introduced the definition of cone monotonicity and given a characterization. See the addendum at the end of this paper for full reference information.


Introduction
Though monotonicity for functions from R to R is familiar in even the most elementary courses in mathematics, there are a variety of definitions in the case of functions from R to R. In this paper we review the definitions we found in the literature and suggest a new definition (with its variants) which we find useful.
In Section 2, we introduce the definitions from the literature for dimensional monotone functions ( ≥ 2).
We give examples and discuss the relationship between these definitions. In Section 3, we introduce a new definition of monotonicity and some of its variants. We then give examples and explore the characteristics of these new definitions.

Definitions and Examples
In [1], Lebesgue notes that on any interval in R a monotonic function attains its maximum and minimum at the endpoints of this interval. This is the motivation he uses to define monotonic functions on an open bounded domain, Ω ⊂ R 2 . His definition requires that these functions must attain their maximum and minimum values on the boundaries of the closed subsets of Ω. We state the definition found in [1] below.
Definition 1 (Lebesgue). Let  Remark 3. Notice also that we can extend this definition to a function : Ω ⊂ R → R.
We now give a couple examples of functions that are Lebesgue monotonic. Example 5. Let Ω = ( , ) be the square of side length , centered at a point ∈ R , for some > 0. Any function of real variables whose level sets are lines is Lebesgue monotone. For example, let ( , ) = 3 − (see Figure 1). Because the function is constant in the direction, we see that on the boundary of any closed subset of Ω, must take on all the same values as it takes in the interior. Of course, the choice of Ω is somewhat arbitrary here (it need only be bounded).
We now move on to another definition given in [2]. Here Mostow gives the following definition for monotone functions.
We see that if Ω = R 2 then we can choose a closed disk, = (0, ) centered at the origin with radius so that = R 2 \ . On = a function, , that is Mostow monotone must obtain both its maximum and its minimum. But, we can let ↘ 0. In doing this, we see that the maximum and minimum of can be arbitrarily close. This tells us that if is Mostow Monotone, then it must be a constant function. In [2], Mostow states that one can adjust this definition by requiring the function to take on its maximum or minimum on only for relatively compact open sets.
Example 7. It is not true that Lebesgue monotone functions are Mostow monotone (even if we follow the suggestion in [2] to adjust the definition of Mostow monotone). To see this, we consider a function : Ω ⊂ R 2 → R that is affine and has its gradient oriented along the domain as in Figure 2. Here will have supremum and infimum that are not attained on the boundary of the open set .
Remark 8. Notice that if Ω ⊂ R 2 is a bounded domain then any continuous, Mostow monotone function is also Lebesgue monotone. This is true whether or not we are adjusting the definition as suggested in [2]. Before giving the next definition, we give some notation for clarity. Let Ω ⊆ R 2 be an open domain, ( , ) the closed ball of radius around the point ∈ Ω, and ( , ) the boundary of the ball, ( , ). We say a function is 1 loc (Ω) if ∫ | | < ∞ for every bounded set ⊂ Ω. For comparison, we write the following definition for a less general function than what can be found in [3].
has measure zero. A function is then said to be Vodopyanov-Goldstein monotone on a domain, Ω, if it is Vodopyanov-Goldstein monotone at each point ∈ Ω.
Example 10. If we remove the continuity requirement for both Lebesgue and Mostow monotone functions we can create a function that is Mostow monotone but not Vodopyanov-Goldstein monotone. For the function in Figure 3, we see that any closed and bounded set must attain both the maximum and minimum of on its boundary, but if we take a ball, , that contains the set { = 0}, we see that ( ) = {−1, 1}. So, −1 ( ( ∩ R \ ( ))) ∩ does not have measure zero. That is, is not Vodopyanov-Goldstein monotone.
Example 11. Now, a function can be Vodopyanov-Goldstein monotone, but not Lebesgue monotone. An example of such a function is one in which attains a minimum along a set, M, that is long and narrow relative to the set Ω (see Figure 4). In this case, the boundary of any ball, ( , ) ⊂ Ω, that is centered along this set must intersect the set, M, thus attaining both its maximum and minimum on the boundary of the ball, but the function will not reach its minimum on the boundary of a closed set Ω such as the one in Figure 4. The next theorem shows that, for continuous functions, Lebesgue monotone functions are Vodopyanov-Goldstein monotone.

Theorem 12. Let Ω ⊂ R 2 be a bounded domain and let
: Ω → R be continuous. Then is Vodopyanov-Goldstein monotone function if is Lebesgue monotone.
Proof. Suppose is Lebesgue monotone, then we know that for all closed sets Ω ⊂ Ω, attains its local extrema on Ω . In particular, if we let ∈ Ω, we have that attains its local extrema on the boundary of ( , ) for any > 0. Let and be such that Then we know that ( ( , )) = ( , ) and ( ( , )) = [ , ]. So Thus, So, the measure of the set is zero. Thus, is Vodopyanov-Goldstein monotone at .
Since was chosen arbitrarily, is Vodopyanov-Goldstein monotone.
In [4], Manfredi gives a definition for weakly monotone functions.
Let Ω an open set in R and : Ω → R be a function in 1, loc (Ω). One says that is weakly monotone if for every relatively compact subdomain Ω ⊂ Ω and for every pair of constants ≤ such that one has that Manfredi also gives the following example of a function that is weakly monotone, but not continuous (in this case at the origin).
Example 14 (Manfredi). Write = for ∈ R 2 . Define by We expect that all the above types of monotone functions should be weakly monotone. Because this function is not continuous, it does not satisfy the definition of Lebesgue or Mostow montone.
Theorem 15. Let Ω ⊂ R 2 be a bounded domain and : Ω → R, if is Lebesgue monotone, then is weakly monotone.
Remark 16. Using Theorem 15 and Remark 8, we see that a function that is Mostow Monotone is also weakly monotone.

Normal Monotone, Cone Monotone, and Monotone
In this section, we introduce a new definition of monotonicity which we call Cone monotone. We will discuss some variants of this new definition that we call Normal monotone and monotone. We also characterize monotone functions.

Cone Monotone.
Motivated by the notion of monotone operators, we give a more general definition of monotonicity for functions in 2 dimensions. But first, we define the partial ordering, ≤ on R 2 .
Definition 17. Given a convex cone, ⊂ R 2 and two points , ∈ R 2 , one says that Definition 18. One says a function : Ω ⊆ R 2 → R is cone monotone if at each ∈ Ω there exists a cone, ( ), so that We say a function is monotone if the the function is cone monotone with a fixed cone .

Characterization of Cone Monotone.
Here we first notice that a function that is monotone cannot have any local extrema. This is stated more precisely in the following theorem.
Theorem 19. Assume is a convex cone with nonempty interior. If is monotone then there is no compact connected set so that ( ) is a local extremum.
Proof. Suppose to the contrary. That is, suppose that ( ) is a local minimum and suppose is monotone. Then we have for every point ∈ and every ∈ ( ) \ , that ( ) < ( ) (see Figure 5).
Remark 20. Theorem 19 and Remark 2 give us that a continuous monotone function is also Lebesgue monotone.
For the following discussion, we work in the graph space, R +1 of a monotone function : R → R. Assume a fixed closed, convex cone, with nonempty interior. Set Let ⃗ denote the vector ( 1 , 2 , . . . , ). We can translate these sections up to the graph of so that it touches at the point ( ⃗ , ( ⃗ )). In doing this we see that we have (see Figure 6)   Care needs to be taken in the case when has a jump discontinuity at ⃗ . Since for example, for an upper semicontinuous function epi does not contain points along the vertical section, {( ⃗ , ) | ≤ ( ⃗ )}, below the point ( ⃗ , ( ⃗ )). Let Using a limiting argument we notice that indeed this vertical section is contained in E. If ( ⃗ , ) ∈ {( ⃗ , ) | ≤ ( ⃗ )}, then we can find a sequence of points, { ⃗ } ⊂ R so that ⃗ → ⃗ . Thus, for large enough, |( ⃗ , ) − ( ⃗ , )| is small. Thus, cl(E) = cl(epi ). A similar argument can be used to give the second equation in (16) for lower semicontinuous. Using these two results, we get that (16) holds for any function .

Examples of Cone Monotone Functions. We now consider some examples of monotone functions.
Suppose is a ray so that has empty interior. Then for to be monotone all we need is for monotone on all lines parallel to , that is monotone in the positive direction of . Therefore, need not even be measurable. but is sloped upward (see Figure 8). More specifically, the function ( , ) = sin( ) + + is monotone. We can see this by noticing that is increasing in the cone Remark 24. Notice in this example that has oscillatory behavior. Yet, is still cone monotone. Notice also that if an oscillating function is tipped enough the result is monotone. The more tipped the more oscillations possible and still be able to maintain monotonicity.
Example 25. Some cone monotone functions are monotone in no other sense. An example of a function, : R 2 → R, that is Cone monotone, but not Vodopyanov-Goldstein monotone, is a function whose graph is a paraboloid. At each point , that is not the vertex of the paraboloid, we find the normal to the level set { = ( )}. We see the half space determined by this normal is a cone in which increases from ( ). At the vertex of the paraboloid, we see that all of R 2 is the cone in which increases.
Example 26. Not all Vodopyanov-Goldstein Monotone functions are cone monotone. An example of a function that is Vodopyanov-Goldstein Monotone, but is not cone monotone, can be constructed with inspiration from Example 11. Level sets of this function are drawn in Figure 9. Here we see that the darkest blue level (minimum) set turns too much to be Cone monotone. We see this at the point . At this point, there is no cone so that all points inside the cone have function value larger than ( ) since any cone will cross the dark blue level set at another point.
Example 27. We can create a function, , that is Lebesgue Monotone, but is not Cone monotone. In this case, we need a level set that turns too much, but the level sets extend to the boundary of Ω. We see such a function in Figure 10. Let dark blue represent a minimum. Then at the point , there is no cone that so that every point in the cone has function value larger than ( ) since every cone will cross the dark blue level set. Now if the domain has dimension higher than 2 and is convex and has empty interior, but is not just a ray, then we can look at slices of the domain (see Figure 11). We can see that on each slice of the domain, the function still satisfies Theorems 19 and 21. But, we also see that the behavior of 6 Abstract and Applied Analysis y Figure 10: A function that is Lebesgue monotone, but is not Cone monotone. Figure 11: Cones with empty interior in a 3D domain that are not just a ray. the function from slice to slice is independent. This is the same behavior as we see when the function is defined on a 2-dimensional domain and is a ray. That is, from line to line, the function behavior is independent (see Example 22). We can also see an example of the extended cones for a monotone function where is a ray, in Figure 12.
If is a closed half space then has level sets that are hyperplanes parallel to and is one dimensional monotone.  We can also construct a monotone function by taking arbitrary unions of the sets ( ⃗ , +1 ) + . By construction the boundary of this set is then the graph of the epigraph (and of the epograph) of a monotone function.

Bounds on TV Norm.
In this section, we find a bound on the total variation of monotone functions. To do this we use the idea of a tipped graph introduced in Section 3.1.1.
Using the coarea formula discussed above, we get an upper bound on the total variation of a function that is monotone as follows: 3.2. Normal Monotone. Motivated by the nondecreasing (or nonincreasing) behavior of monotone functions with domain in R, we introduce a specific case of cone monotone. We consider a notion of monotonicity for functions whose domain is in Ω ⊂ R 2 by requiring that a monotone function be nondecreasing (or nonincreasing) in a direction normal to the level sets of the function. First, we introduce a few definitions.
Definition 28. One says that a vector V is tangent to a set at a point ∈ if there is a sequence { } ⊂ with → and a sequence { } ⊂ R with ↘ 0 so that The set of all tangents to the set at the point ∈ is the tangent cone and denote it by ( ).
Abstract and Applied Analysis 7 K Ω Figure 13: A function that is monotone, but not Normal monotone.
Definition 29. One says that a vector ( ) is normal to a set at a point ∈ if for every vector V ∈ ( ), ( ) ⋅ V ≤ 0.
Definition 30. One says that a function : R 2 → R is (strictly) Normal monotone if for every ∈ R and every on the boundary of the level set { = } the 1-dimensional functions → ( + ( )) are (strictly) monotone for every vector, ( ), normal to the level set { = } at .
Remark 31. The definition for normal monotone requires that the function be monotone along the entire intersection of a one dimensional line and the the domain of . In the case of cone monotone, we require only monotonicity in the direction of the positive cone while in the case of monotone, the fact that we can look forwards and backwards to get nondecreasing and nonincreasing behavior follows from the invariance of , not the definition of cone monotone.
Remark 32. Notice also that Example 23 is not normal monotone.
Remark 33. A smooth function that is normal monotone is cone monotone for any cone valued function ( ) ⊂ ( ) for all . We now explore this definition with a few examples.
Example 34. One can easily verify that a function whose graph is a non-horizontal plane is strictly normal monotone. This is desirable since a 1D function whose graph is a line is strictly monotone (assuming it is not constant).
Example 35. A function whose graph is a parabola is not monotone in 1D so neither should a paraboloid be normal monotone. One can easily verify this to be the case.
Example 36. If we extend a nonmonotone 1D function to 2D, we should get a function that is not normal monotone. An example of such a function is the function ( , ) = 3 − . Notice, this function is Lebesgue monotone, but neither nor Normal monotone.
Example 37. In Figure 13, we show a function whose level sets are very oscillatory so that it is not normal monotone, while still being monotone. Ω x y Figure 14: A function that is not monotone, but is Normal monotone.
Example 38. In Figure 14, we see that if Ω is not convex, then we can construct a function that is not monotone, but is Normal monotone. In this example, the function increases in a counter clockwise direction. This function is Normal monotone. We can see that it is not monotone since at the point any direction pointing to the north and west of the level line is a direction of ascent. But, at the point , these directions are directions of descent. So the only cone of ascent at both and must be along the line parallel to their level curves. But, we see that at other points, this is not a cone of ascent. Thus, this function cannot be monotone.
The next theorem tells us that a normal monotone function is also Lebesgue monotone.
Theorem 39. Let Ω ⊂ R 2 be a bounded domain and let : Ω → R be a continuous, normal monotone function then is also Lebesgue monotone.
Proof. We prove the contrapositive. Suppose is not Lebesgue Monotone. Then there exists a set Ω so that We want to show that is not normal monotone. Let us then define the nonempty set ⊂ Ω to be the set where attains a local minimum, at every point in . That is, Let ( 0 , 0 ) ∈ and let ( 0 , 0 ) be a normal at ( 0 , 0 ) to . We know then that → (( 0 , 0 ) + ( 0 , 0 )) is not monotone since has a local minimum on . Thus is not normal monotone.
Remark 40. This theorem gives us that a function that is normal monotone is also weakly monotone.  Figure 15: Types of monotonicity in higher dimensions and how they compare for a continuous function.

Conclusion
In this paper we explored current and new definitions of monotone. For continuous functions, we compared several definitions of monotonicity, in higher dimensions. How these sets of functions are related is represented in the Venn diagram in Figure 15.
We also showed how to construct monotone functions. We show that bounded monotone functions are BV and we find a bound on the total variation of these functions.