Ostrowski type inequalities for sets and functions of bounded variation

In this paper we obtain sharp Ostrowski type inequalities for multidimensional sets of bounded variation and multivariate functions of bounded variation.


Introduction
In  Ostrowski [] proved the following inequality. The inequality is sharp.
Inequalities that estimate deviation of a function from its mean value using different characteristics of the function are usually called Ostrowski type inequalities. Such inequalities have many applications, in particular in the area of numerical methods, and are heavily studied. See [] and the references therein for results connected with Ostrowski type inequalities for univariate functions of bounded variation and their applications.
The goal of this article is to obtain sharp Ostrowski type inequalities for multivariate functions and multidimensional sets of bounded variations. There are several ways to extend the notion of bounded variation to multivariate functions, see [] for a review of different approaches for functions of two variables; [] for the point of view that is generally accepted in literature now.
We introduce a new definition of bounded variation that is based on the Kronrod-Vitushkin approach []. The introduced variation of a multivariate function has (unlike any of the Kronrod-Vitushkin variations) the following two properties: the variation does not change if the argument of the function is multiplied by a non-zero constant; and the variation of a multivariate radial function is twice bigger than the variation of the generating one-dimensional function, see Properties  and  below for rigorous statements of the properties.
The paper is organized as follows. In Section  we list the notations used throughout the paper. In Sections  and  we introduce definitions, justify the correctness of the definitions, and list some properties of the sets and function variations. Section  is devoted to Ostrowski type inequalities.

Notations
For a set F ⊂ R d , denote by ∂F, int F and F its boundary, interior and closure, respectively. For arbitrary t ∈ R, set R d t := {(x, t) ∈ R d : x ∈ R d- }. For x, y ∈ R d , by xy we denote the segment with ends in the points x and y, i.e., xy = {(t)x + ty : t ∈ [, ]}. For c ∈ R and where |w| denotes the Euclidean distance between the point w ∈ R d and zero element θ of R d . For For an arbitrary function f : By P d- we denote the d - dimensional real projective space, i.e., the set of all lines in R d that contain θ . The distance between two lines r  , r  ∈ P d- is by definition equal to the angle between r  and r  . The measure of a set A ⊂ P d- is by definition equal to the spherical measure of the set l∈A l ∩ S d- ; so that the measure of P d- is equal to the measure of S d- .
For each r ∈ P d- , by d- (r) we denote the hyperplane that contains θ and is orthogonal to the line r; d- (r) is considered as a d --dimensional space with d --dimensional Lebesgue measure and Euclidean metric. For each β ∈ d- (r), by l(r, β) we denote the line that contains β and is parallel to r.
It is assumed that product topology is induced on the Cartesian product of a finite number of topological spaces and product measure is induced on the Cartesian product of a finite number of measure spaces. By μ k , k ∈ N, we denote the k-dimensional Lebesgue measure in R k ; by μ we denote the spherical measure on the unit sphere S d- and the measure on the projective space P d- .

Definition
Definition  Denote by N(F) the number of connected components of the set F ⊂ R d ;  for an empty set, and +∞ if the set of connected components is infinite.
Variation of a set F determined by r ∈ P d- is defined by the following formula.
Definition  For a compact set F ⊂ R d and a line r ∈ P d- , set v(F, r) := ess sup where the supremum is taken over all partitions F = n k= F k of the set F into a finite number of compact pairwise ε-disjoint subsets F k .
In this case the supremum in the definition is taken over all partitions F = n k= F k of the set F into a finite number of compact disjoint subsets F k .

Correctness of the definitions
The proofs of measurability of the functions that stay under integral signs (Lemmas -) use ideas from [] (Chapters -) and [] (Chapter ).
Throughout this subsection, we identify each point (r, x) of the space P d- × R d- (r ∈ P d- , x ∈ R d- ) with the line l(r, β), where β ∈ d- (r) is a point with coordinates x with respect to some orthonormal basis of d- (r) (the basis of d- (r) is assumed to continuously change as r ∈ P d- changes).
We need the following lemma.
Denote by ψ  one of continuous maps from [, ] onto S d- ; such a map exists due to the Hahn-Mazurkiewicz theorem; see, for example, Theorem . in []. Define a function ψ  : [, ] × R d → P d- × R d- using the following rule. For each (t, y) ∈ [, ] × R d , let ψ  (t, y) be the line l(r, β), where r ∈ P d- is the line that contains ψ  (t), and β is such that the line l(r, β) contains the point y. It is easy to see that the function ψ  is continuous. Then ψ(B) = ψ  (B), whereB := [, ] × B. This means that ψ(B) is a continuous image of a Borel setB ⊂ R d+ and hence is measurable (see, for example, Theorem  in []).
The set ϕ(B) is a projection of the Borel set B to the t-axis of R d , hence is measurable.
Definition  For a compact set F ⊂ R d and ε > , denote by N ε (F) the number of εcomponents of the set F;  for an empty set.

Lemma  Let a compact set F ⊂ R d be given. Then, for arbitrary
With each ε-component W of the set F, associate a ball with center in an arbitrary point w ∈ W and radius ε  . Balls that correspond to different ε-components of F are pairwise disjoint; hence there can be only a finite number of such balls due to the boundedness of F.
If F has a finite number of connected components F  , . . . , F n , then ε  := min i =j ρ(F i , F j ) >  due to the compactness of components of a compact set. and The following lemma holds.
Lemma  Let a compact set F ⊂ R d and ε >  be given. The function v ε F is measurable Consider the (countable) set of all closed balls in R d with rational centers and radii. Let˜ be the set of all finite unions of the balls from . For each n ∈ N, define n to be the family of all sets of the form m s= B i s , where m ≥ n, and {B i s } m s= is a collection of pairwise ε-disjoint sets from˜ . Then the set n is countable for each n ∈ N.
Note that the functions v ε F and N ε F take only non-negative integer values. The sets {v ε and where the functions ψ and ϕ are defined in Lemma . We prove equality (); the other one can be proved using similar arguments.
If (r, x) belongs to the right-hand side of (), then there exists a set B = m s= B i s ∈ n such that l(r, β) ∩ B i s ∩ F is a non-empty set strictly inside B i s , s = , . . . , m. Since the sets that constitute B are ε-disjoint, by the definition of n , we obtain that v ε Consider a finite cover C of the compact set l(r, β) ∩ F by open balls with rational centers and radii such that each ball has diameter less than δ and contains some point from l(r, β) ∩ F. Denote by B k the union of closures of all balls from the cover C that intersect F k , k = , . . . , m. Then m k= B k belongs to n by construction, and hence (r, x) belongs to the right-hand side of ().
Since F is a closed set and n is a countable set, Lemma  implies that the sets {v ε Tonelli's theorem and Lemma  imply that the function v p (F) is well defined for every  ≤ p ≤ ∞ and compact F ⊂ R d ; hence the functions V ε p (F) are also well defined for all ε ≥ .

Some properties of the sets variation
The following property is a direct consequence of the definitions.
This implies (). The case when V ε p (F) = ∞ can be considered in a similar way.
Property  Let n ∈ N and pairwise disjoint compact sets F  , . . . , F n ⊂ R d be given. Then, It is sufficient to prove the property in the case n = . Set F := F  ∪ F  .
Since F  and F  are compact disjoint sets, we have ρ(F  , F  ) > , and hence, for arbitrary r ∈ P d- and β ∈ d- (r), one has N(F ∩ l(r, β)) = N(F  ∩ l(r, β)) + N(F  ∩ l(r, β)). This Property  Assume n ∈ N, ε ≥  and pairwise ε-disjoint compact sets F  , . . . , F n ⊂ R d are given. Then, for all p ∈ [, ∞], It is sufficient to prove the property in the case n = .
be a partition of the set F into compact pairwise ε-disjoint subsets. Then, by Property , On the other hand, for arbitrary partitions of the sets F  and F  into compact ε-disjoint sets {T  k } s k= and {T  k } m k= respectively, s, m ∈ N, is a partition of the set F into compact ε-disjoint sets, and hence Property  If ε ≥  and F ⊂ R d is a compact set that has exactly n ∈ N ε-connected components F  , . . . , F n , then, for all p ∈ [, ∞], V ε p (F) = n k= v p (F k ).
Note that each ε-connected component of a compact set is compact. Hence, by Properties  and , V ε . The property follows from the observation that for arbitrary r ∈ P d- and β ∈ d- (r) one has N(F ∩ l(r, β)) = N(αF ∩ l(r, αβ)) and hence v(αF, r) = v(F, r).

Variation of a function 4.1 Definition
Definition  Let a set E ⊂ R d and a function f : E → R be given. For t ∈ R, the set is called a level set of the function f .
The variation of a continuous function is given by the following definition.
If F is locally connected, then set

Correctness of the definitions
We need to prove that the functions under the integral signs are measurable. Without loss of generality, we may assume that R d ⊃ F = E, and we need to prove that the function v p (L(f ; ·)) is measurable. Consider the graph of the function f and two functions v (f ) : Since the set E is compact and the function f is continuous on E, the set (f ) ⊂ R d+ is compact. This, by Lemma , implies that the function v (f ) is measurable (P d × R d ).
Recall that R d+ that maps a point (t, r, x) to the point (R, y) with R = {(z, ), z ∈ r} and y = (x, t) is continuous and has continuous inverse. Moreover, for arbitrary (t, r, , r, x)). Hence, for arbitrary c ∈ R, the function φ maps the set The latter is an intersection of a measurable (due to measurability of v (f ) ) set and a closed set; hence the former is also a measurable set. This means that the function v f is mea- Without loss of generalization, we may assume that F = E. Taking into account Property , it is sufficient to prove that each of the functions V ν ). Due to Lemma , for each k = , , . . . , the set T k := {N f = k} \ T extr is measurable; moreover, obviously these sets are pairwise disjoint.
Let c ∈ R be given. Then and it is sufficient to prove that for each k ∈ N the set {t ∈ T k : V ν p (L(f ; t)) ≤ c} is measurable.
Let some k ∈ N and t * ∈ T k be fixed. The set L(f ; t * ) contains exactly k ν-connected components F  , . . . , F k . Each of the components is a compact set, hence there exists ε >  such that the sets U s (ε) = {x : ρ(x, F s ) < ε}, s = , . . . , k, are pairwise ν-disjoint. Set U(ε) := k s= U s (ε). There exists δ >  such that L(f ; t) ⊂ U(ε) for all t ∈ (t *δ, t * + δ). Really, assume the contrary, suppose that there exists a sequence a n , a n →  as n → ∞, such that each of the level sets L(f ; t * + a n ) contains a point x n / ∈ U(ε). Switching to a subsequence, if needed, we may assume that the sequence x n converges to some x ∈ R d . Since U(ε) is an open set, x / ∈ U(ε). However, this is impossible since f is continuous, and hence f (x) = t * .
For each s = , . . . , k, consider an arbitrary point x s ∈ F s . Since the level set L(f , t * ) does not contain extremums, F is locally connected and f is continuous; for small enough ε s > , the set f (B d (x s , ε s )) contains a neighborhood (t *δ s , t * + δ s ) of t * (δ s > ). Hence, for arbitrary t ∈ (t *δ s , t * + δ s ), the level set L(f ; t) contains at least one ν-component inside U s (ε). This means that there exists δ = δ(t * ) >  such that each of the level sets L(f , t), t ∈ (t *δ, t * + δ) contains at least one ν-component inside U s (ε), s = , . . . , k. Hence, for all t ∈ T k ∩ (t *δ, t * + δ), the level set L(f ; t) contains exactly one ν-component inside U s (ε), s = , . . . , k. This implies that for all t ∈ T k ∩ (t *δ, t * + δ) due to Property . For each t * ∈ T k , set W (t * ) := (t *δ(t * ), t * + δ(t * )); the sets W (t * ), t * ∈ T k , constitute an open cover of the set T k . Since R is a Lindelöf space, we can find a countable subcover We obtain a countable partition of the set T k into pairwise disjoint measurable subsetsW m , m ∈ N, such that on each of the setsW m we have representation () of the function V ν p (L(f ; ·)) (the sets U s (ε), s = , . . . , k, might be different for different m ∈ N). Due to Lemma , this implies the measurability of the set and hence the lemma is proved.

Some properties of the function variation
Below we list some properties of the function variation. Everywhere p ∈ [, ∞] and a compact set F ⊂ R d are fixed, f is continuous on F function. For properties of V p (f ), the set F is further assumed to be locally connected.
The fact that variations are non-negative follows from the definition. If f is constant, then it has exactly one non-empty level set.

Property  For arbitrary
The property follows from Property .
Property  Let t ∈ R be such that F \ L(f ; t) has exactly n ≥  connected components F  , F  , . . . , F n . Assume that for all k = , . . . , n, F k \ F k ⊂ L(f ; t). Then v p (f ; F) ≤ n k= v p (f ; F k ) and V p (f ; F) = n k= V p (f ; F k ).
Consider arbitrary s ∈ R, s = t. For k = , . . . , n, set W k := F k ∩ L(f ; s). Then the set W k is closed, k = , . . . , n, and W k ⊂ F k due to conditions of the property and the fact that different level sets of any function are disjoint. This means that W k are compact pairwise disjoint sets. From Properties  and  it follows that V p (L(f ; s)) = n k= V p (F k ∩ L(f ; s)) and v p (L(f ; s)) ≤ n k= v p (F k ∩ L(f ; s)). The statement of the property now follows from Definition .

Property 
In the case d = , the property follows from Property , so we can assume that d ≥ . Let arbitrary t = ϕ() be fixed. For arbitrary r ∈ P d- and β ∈ d- (r) the number N(L(f ϕ ; t) ∩ l(r, β)) can be obtained by the following procedure: consider the line r = l(r, θ ) and mark points of the set L(f ϕ ; t)∩l(r, θ ); cut the interval (-|β|, |β|) from the line and stick the points -|β| and |β| together; the number of components of marked points on the obtained 'cut' line is equal to N(L(f ϕ ; t) ∩ l(r, β)). This shows that for arbitrary β From the choice of t it follows that and hence there exists ε >  such that B(ε) ∩ L(f ϕ ; t) = ∅. This implies that the set L(f ϕ ; t) ∩ l(r, θ ) does not contain points x with |x| < ε and hence for all β such that |β| < ε () becomes equality. This implies that v(L(f ϕ ; t), r) = N(L(f ϕ ; t) ∩ l(r, θ )). From () it follows that N(L(f ϕ ; t) ∩ l(r, θ )) =  · N(L(ϕ; t)). Equality v p (f ϕ ; B) = ·   ϕ follows from Property  now. Equality v p (f ϕ ; B) = V p (f ϕ ; B) follows from the geometry of the level sets of f ϕ .

Auxiliary results
Lemma  Let d ∈ N, d ≥ , ε > , x ∈ R d , r ∈ P d- and a measurable set F ⊂ B d (x, ε) be given. For arbitrary A ∈ (, ), there exists α = α(A) ∈ (, ) that does not depend on ε, x and r such that The fact that α does not depend on ε follows from the observation that The fact that α is independent of x and r is obvious. The existence of α follows from the equality where y ∈ d- (r) is such that the line l(r, y) contains x and equality μ d- Lemma  Let p ∈ [, ∞), A >  and B ∈ [, A] be given. Then It is sufficient to prove that the function ϕ(x) =  p + ( - p )x -(x) p is non-negative on [, ]. Since ϕ() = ϕ() = , the function ϕ has at least one zero on (, ). The function ϕ (x) = p(x) p- +  - p is decreasing on [, ], hence has at most one zero on (, ). This implies that ϕ () >  and hence the function ϕ is increasing on [, x * ] and is decreasing on [x * , ], where x * is zero of ϕ on (, ); hence ϕ is non-negative on [, ].

Main results
The following theorem is the main tool to prove Ostrowski type inequalities for functions and sets of bounded variation below. Choose n so big that ρ n < δ. Then due to () and (). Moreover, since x, y ∈ l(r, β), we receive that (   ) Set =  (ρ n ) ∩  (ρ n ). Then, due to (), () and (), μ d- > . But each line l(r, β), β ∈ , contains a point from W due to Condition  of Theorem  and the definitions of the sets  (ρ n ) and  (ρ n ); this contradicts assumption v(W , r) =  of the lemma.
Lemma  Assume that the conditions of Theorem  hold. Let R ⊂ P d- be such that v(W , r) =  for all r ∈ R. If R contains d lines that are not contained in any d --dimensional hyperplane, then μ d (F) = .
Due to () it is enough to prove thatF = ∅. Let r  , . . . , r d be the lines from the statement of the lemma, and let ρ  , . . . , ρ d be unit vectors parallel to these lines. Set P := { d k= t k ρ k : t k ∈ (-, ), k = , . . . , d}, then P is an open in R d set. Consider arbitrary x ∈ int B d . Choose ε >  such that x + εP ⊂ B d . Then, for all points y from the segment θ x, P y := y + εP ⊂ B d . y∈θx P y is an open cover of a compact set θ x, hence it contains a finite subcover P  , P  , . . . , P m , m ∈ N. From Lemma  it follows that for each s = , . . . , m either P s ⊂F, or P s ∩F = ∅.

Proof of Theorem 1
If v(W , r) ≥  for almost all r ∈ P d- , then v p (W ) ≥  and inequality () holds. It is strict because Condition  of Theorem  holds. If there is a set R ⊂ P d- of positive measure such that v(W , r) =  for all r ∈ R, then μ d F =  due to Lemma , and inequality () holds. Assume that there exists R ⊂ P d- , μR > , such that v(W , r) =  for all r ∈ R and v(W , r) ≥  for almost all r ∈ P d- \ R. Then