Metric Fourier approximation of set-valued functions of bounded variation

We introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

spline operator, the polynomial interpolation operator. While in older papers the approximated SVFs are mainly continuous, the later works [23,13] are concerned with multifunctions of bounded variation.
The main topic is an adaptation of the trigonometric Fourier series to set-valued functions of bounded variation with general compact images. We also try to obtain error bounds under minimal regularity requirements on the multifunctions to be approximated and focus on the investigation on SVFs of bounded variation. We use in our analysis some properties of maps of bounded variation with values in metric spaces proved in [15].
We are familiar only with few works on trigonometric approximation of multifunctions. Some results on this topic for convex-valued SVFs by methods based on the Aumann integral are obtained in [6]. For the related topic of trigonometric approximation of fuzzy-valued functions see, e.g. [2,12,38,25]. Note that in this context the level sets determine multifunctions with convex values (intervals in R).
In this paper we define the metric analogue of the partial sums of the Fourier series of a multifunction via convolutions with the Dirichlet kernel of order n, for n ≥ 0, the convolutions being defined as weighted metric integrals. To study error bounds of these approximants and to prove convergence as n → ∞, we introduce new one-sided local moduli of continuity in Section 3 and quasi-moduli of continuity in Section 6. The main result of the paper is analogous to the classical Dirichlet-Jordan Theorem for real functions [39]. It states the pointwise convergence in the Hausdorff metric of the metric Fourier approximants of a multifunction of bounded variation to a compact set. In particular, if the multifunction F is of bounded variation and continuous at a point x, then the metric Fourier approximants of it at x converge to F (x). The convergence is uniform in closed finite intervals where F is continuous. At a point of discontinuity the limit set is determined by the values of the metric selections of F there.
The paper is organized as follows. In the next section some basic notions and notation are recalled. Onesided local moduli of continuity of univariate functions with values in a metric space are introduced and studied in Section 3. The theory developed in Section 3 is specified in Section 4 to set-valued functions of bounded variation, to their chain functions and metric selections. In Section 5 the weighted metric integral is introduced and some of its properties are derived. The main results of the paper are presented in Section 6. To make the reading easier, the section is divided into three subsections. The first subsection contains the definition of the metric Fourier approximants of multifunctions. The second subsection contains a refinement of the classical Dirichlet-Jordan Theorem [39]. There we obtain error bounds for the Fourier approximants for special classes of real functions of bounded variation. This refinement is used in the third subsection for the main results on the metric Fourier approximation of set-valued functions. In Section 7 we discuss properties of a set-valued function and of its metric selections at a point of discontinuity and study the structure of the limit set of the metric Fourier approximants.
There are two appendices: Appendix A contains the proof of Theorem 4.13 which is stated without a proof in Section 4 of [23]. Appendix B contains the proof of the refined Dirichlet-Jordan Theorem from Subsection 6.2.

Preliminaries
In this section we introduce some notation and basic notions related to sets and set-valued functions.
All sets considered from now on are sets in R d . We denote by K(R d ) the collection of all compact non-empty subsets of R d . By Co(R d ) we denote the collection of all convex sets in K(R d ). The convex hull of a set A is denoted by co(A). The metric in R d is of the form ρ(u, v) = |u − v|, where | · | is a norm on R d . Note that all norms on R d are equivalent. In the following we fix one norm in R d . Recall that R d is a complete metric space.
Let A and B be non-empty subsets of R d . To measure the distance between A and B, we use the Hausdorff metric based on ρ where the distance from a point c to a set D is dist(c, D)ρ = inf d∈D ρ(c, d).
It is well known that K(R d ) and Co(R d ) are complete metric spaces with respect to the Hausdorff metric [33,35]. For an arbitrary metric space (X, ρ), the same formula (1) defines a metric on the set C(X) of all non-empty closed subsets of X. It is known that the metric space (C(X), haus) is complete if (X, ρ) is complete. Moreover, (C(X), haus) is compact if X is compact (e.g. [1,Section 4.4]).
We denote by |A| = haus(A, {0}) the "norm" of the set A ∈ K(R d ). In [23], the three last-named authors introduced the notions of a metric chain and of a metric linear combination as follows.
Note that the metric linear combination depends on the order of the sets, in contrast to the Minkowski linear combination of sets which is defined by For a sequence of sets {An} ∞ n=1 the lower Kuratowski limit is the set of all limit points of converging sequences {an} ∞ n=1 , where an ∈ An, namely, lim inf n→∞ An = a : ∃ an ∈ An such that lim n→∞ an = a .
Analogously, for a set-valued function F : , k ∈ N, and y k → y} .
The upper Kuratowski limit is the set of all limit points of converging subsequences {an k } ∞ k=1 , where an k ∈ An k , k ∈ N, namely lim sup n→∞ An = a : ∃ {n k } ∞ k=1 , n k+1 > n k , k ∈ N, ∃ an k ∈ An k such that lim k→∞ an k = a .
Correspondingly, for a set-valued function F :

Regularity measures of functions with values in a metric space
Here we consider regularity measures of functions defined on a fixed interval [a, b] ⊂ R with values in a complete metric space (X, ρ). A basic notion in this paper is the modulus-bounding function ω(δ) which is a non-decreasing function ω : [0, ∞) → [0, ∞). Frequently we occur the situation when in addition lim δ→0 + ω(δ) = 0, but we do not require this in the definition.
In the analysis of continuity of a function at a point, the notion of the local modulus of continuity is instrumental [34] To characterize left and right continuity of functions, we introduce the left and the right local moduli of continuity, respectively.
Similarly, the right local modulus of continuity of f at Remark 3.2.
(i) One can define the one-sided local moduli of continuity analogously to (2), for example, the left local modulus as Yet it is easily seen that this quantity is equivalent to (3), namely (ii) Note that the classical global modulus of continuity ω f, δ = sup this property is not satisfied by the local moduli.
The following relations hold for x * ∈ [a, b]: In the next proposition we extend some properties known for the local modulus of continuity ω(f, x * , δ) to the one-sided local moduli.
The function f is right continuous at x * ∈ [a, b) if and only if lim Proof. A function f is left continuous at x * if and only if for every ε > 0 there exists δ > 0 such that Since, by definition, ω − f, x * , δ is non-increasing in δ, the above is equivalent to lim We recall the notion of the variation of a function f : [a, b] → X. Let χ = {x0, . . . , xn}, a = x0 < · · · < xn = b, be a partition of the interval [a, b] with the norm The variation of f on the partition χ is defined as where the supremum is taken over all partitions χ of [a, b]. A function f is said to be of bounded variation if V b a (f ) < ∞. We call functions of bounded variation BV functions and write f ∈ BV[a, b]. If f is also continuous, we write f ∈ CBV [a, b].
and that v f is monotone non-decreasing.
Proof. We prove only the first inequality, the proof of the second one is similar. If x * = a then both sides of the inequality are zero and the claim follows. For x * ∈ (a, b] we have The following claim is a slight refinement of Proposition 1.1.1 in [22] and of [26, Chapter 9, Sec. 32, Theorem 3].
Proof. We prove only the first statement, the proof of the second one is similar.
If v f is left continuous at x * , then by Propositions 3.4 and 3.3 also f is left continuous at x * . Now we prove the other direction. We closely follow the proof in [26, Chapter 9, Sec. 32, Theorem 3].
Assume that f is left continuous at x * . Then for each ε > 0 there exists δ > 0 such that The definition of the total variation implies that one can choose a partition χ = {a = x0 < x1 < · · · < xn = x * } such that Adding more points to χ if necessary, we can guarantee that 0 < x * − xn−1 < δ. Then by (6) we have and Analogs of Propositions 3.4 and 3.5 for the two-sided local modulus of continuity are well-known: Moreover, f is continuous at x * ∈ [a, b] if and only if v f is continuous at x * .
The first statement can be proved along the same lines, and the second statement follows immediately from Proposition 3.5.
Remark 3.7. Note that, in general, ω(f, x * , δ) and ω(v f , x * , δ) are not equivalent for f ∈ BV[a, b]. As an example, consider f (x) = x 2 sin 1 x ∈ BV[0, 1] (where we define f (0) = 0 by continuity). It is easy to see that To estimate the local variation of f , consider the points 1 x k = π 2 + πk, k ∈ N, so that sin 1 Then Helly's Selection Principle (see, e.g. [26,Chapter 6]) will be heavily used in our analysis. We cite a version of it which is relevant to our paper. Helly's Selection Principle. Let {fn} n∈N be a sequence of functions fn : [a, b] → R, and assume that there are constants A, B > 0 such that |fn(x)| ≤ A, n ∈ N, x ∈ [a, b] and V b a (fn) ≤ B, n ∈ N. Then {fn} n∈N contains a subsequence {fn k } k∈N that converges pointwisely to a function f ∞ : In the following statements we consider pointwise limits of sequences of BV functions. We show that the limit function inherits local properties which are shared by the members of the sequence. The first result is known and is given here for the readers' convenience.
In the next theorem we study sequences of functions which are equicontinuous from the left or from the right at a point.
In particular, if lim By the assumption, Let ε > 0 be arbitrarily small. There exists N (ε, z) such that Since ε > 0 was taken arbitrarily, it follows that In particular, it follows from Proposition 3.3 that f ∞ is left continuous at x * .
An analogous result holds for the right continuity at x * .
Arguing along the same lines, one can also prove an analogous statement for the two-sided local modulus of continuity.
In particular, if lim As the last statement in this section, we formulate a property similar to Theorem 3.9 for the local moduli of the function v f .
In particular, if lim By Theorem 3.8 and by the monotonicity of the variation function we have Taking supremum over we get the first claim. The second claim follows from Proposition 3.3.
Analogous statements hold for the right local modulus of continuity and for the two-sided local modulus of continuity.

Multifunctions, their chain functions and metric selections
The main object of this paper are set-valued functions (SVFs, multifunctions) mapping [a, b] to K(R d ). First we recall some basic notions on such SVFs.
The graph of a multifunction F is the set of points in R d+1 defined as It is easy to see that if F ∈ BV[a, b] then Graph(F ) is a bounded set and F has a bounded range, namely We denote the class of SVFs of bounded variation with compact graphs by F[a, b].

For a set-valued function
Below we present some definitions and results from [23] that will be used in this paper. In particular, we recall the definitions of chain functions and metric selections.
Note that the definitions of chain functions and metric selections imply that a metric selection s of a multifunction F is constant in any open interval where the graph of s stays in the interior of Graph(F ).
Through any point α ∈ Graph(F ) there exists a metric selection which we denote by sα. Moreover, F has a representation by metric selections, namely The next statements focus on local regularity properties of chain functions and metric selections. They refine results in [22] and [23].
and let c χ,φ be a chain function corresponding to a partition χ and a metric chain φ as in (7). Then for any Proof. The claim holds trivially for x * = a. So we assume that Otherwise there is i < k such that xi ≤ z < xi+1. By the definitions of the chain function and of the metric chain we get Using the definitions of the variation of F , of vF and of ω − , we continue the estimate: Taking the supremum over z ∈ [x * − δ, x * ] ∩ [a, b] we obtain the claim of the lemma.
Proof. If x * = b, then the claim holds trivially. So we assume that By the definition of the chain function we get Using the definitions of the variation of F , of the variation function vF and (2), (3), (4), (5), we obtain The claim of the lemma follows by taking the supremum over z ∈ [x * , Lemma 4.6. Let F ∈ F[a, b] and let c χ,φ be a chain function corresponding to a partition χ and a metric chain φ. Then for any The above inequalities hold also for x < z = xn. In the case when x = z this estimate is trivial. Taking the supremum over Then Moreover, if F is left continuous at x * then by Propositions 3.5 and 3.
The latter implies that s is left continuous at x * .
Using Lemma 4.5 instead of Lemma 4.4 and arguing as above, we obtain Then Similarly, Lemma 4.6 and Theorem 3.10 lead to Then In particular, if F is continuous at x * , then s is continuous at x * .
Remark 4.10. Analysing the proofs, it is not difficult to see that the estimates in Theorems 4.7-4.9 can be improved in the following way with an arbitrarily small ε > 0. Taking the supremum of the both sides of the last inequality over and ω(vF , δ) is continuous in δ. Taking the limit as ε → 0+ we get ω s, δ ≤ ω vF , δ .
Lemma 4.11. Let F ∈ F[a, b] and let c χ,φ be a chain function corresponding to a partition χ and a metric chain φ. Let δ > 0 be such that Proof. Let χ = {x0, . . . , xn}, a = x0 < · · · < xn = b. By definition, (c χ,φ (xj), c χ,φ (xj+1)) ∈ Π F (xj), F (xj+1) , j = 0, . . . , n − 1. Thus, V x k For the second inequality, we continue the estimate as follows: Theorem 4.12. Let F ∈ F[a, b] and let s be a metric selection of F . Then for all small δ > 0 and all x, 4δ) . Proof. Since s is a metric selection, there exists a sequence of partitions {χn} n∈N with |χn| → 0, n → ∞, and a corresponding sequence of chain functions {cn} n∈N such that s(x) = lim n→∞ cn(x) pointwisely. Take n so large that |χn| < δ, then by Lemma 4.11 we have V x+δ The statement of Theorem 4.12 can be improved in the same manner like in Remark 4.10. Namely, the estimate The next result was announced in [23, Lemma 3.9] without a detailed proof. Although the result is intuitively clear, its proof is rather complicated. We present the full proof in Appendix A.

Weighted metric integral
The well-known Aumann integral [5] of a multifunction F is defined as where A ∈ K(R d ) and The metric integral of SVFs has been introduced in [23]. In contrast to the Aumann integral, the metric integral is free of the undesired effect of the convexification. We recall its definition. First we define the metric Riemann sums. For a multifunction F : [a, b] → K(R d ) and for a partition χ = {x0, . . . , xn}, a = x0 < x1 < · · · < xn = b, the metric Riemann sum of F is defined by The upper limit here is understood in the following sense: y ∈ lim sup |χ|→0 (M)SχF if there is a sequence of partitions {χn} n∈N with |χn| → 0, n → ∞, and a sequence {yn} n∈N such that yn ∈ (M)Sχ n F and yn → y, n → ∞.
It is easy to see that the set (M) b a F (x)dx is non-empty if F has a bounded range. The following result from [23] relates the metric integral of F ∈ F [a, b] to its metric selections.
In this section we define an extension of the metric integral, namely, the weighted metric integral. For a set-valued function F : [a, b] → K(R d ), a weight function k : [a, b] → R and for a partition χ = {x0, . . . , xn}, a = x0 < x1 < · · · < xn = b, we define the weighted metric Riemann sum of F by We define the weighted metric integral of F as the Kuratowski upper limit of weighted metric Riemann sums.
Definition 5.4. The weighted metric integral of F with the weight function k is defined by The set (M k ) b a k(x)F (x)dx is non-empty whenever the SVF kF has a bounded range. Observe that the weighted metric integral of F with the weight k is not the metric integral of the multifunction kF . The difference is that the metric chains in Definition 5.4 are constructed on the base of the function F , and not kF which would be in the latter case.
In the remaining part of this section we extend results obtained for the metric integral in [23] to the weighted metric integral.
Remark 5.5. It is possible to define a "right" weighted metric Riemann sum as and a corresponding weighted metric integral. For BV functions F and k, this integral is identical with This can be concluded from the following lemma. Lemma 5.6. Let F, k ∈ BV[a, b]. Then Proof. Fix a partition χ and consider a corresponding chain φ = (y0, . . . , yn) ∈ CH(F (x0), . . . , F (xn)). We have (F (x0), . . . , F (xn)) .
The next theorem is an extension of Result 5.2 to the weighted metric integral.
Proof. By Result 4.3, every metric selection s of F ∈ F [a, b] is BV, and thus ks is Riemann integrable. Denote Let s be a metric selection of F . Then s is the pointwise limit of a sequence of chain functions {cn} n∈N corresponding to partitions {χn} n∈N with limn→∞ |χn| = 0. Denote kn = kχ n (see (10)) and σn = b a kn(x)cn(x)dx. By Remark 5.3, σn ∈ (M k )Sχ n F . Clearly, kn ∞ ≤ k ∞ and V b a (kn) ≤ V b a (k). By Helly's Selection Principle there exists a subsequence {kn } ∈N that converges pointwisely to a certain function k * . For simplicity we denote this sequence by {kn} n∈N again. It is easy to see that k * (x) = k(x) at all points of continuity of k. Indeed, for a partition χn there is an index in such that x ∈ [xi n , xi n+1 ), where xi n and xi n +1 are subsequent points in χn. By (10) we get

It remains to show the converse inclusion
Moreover, Corollary 5.8 implies the following "inclusion property" of the weighted metric integral as stated below.
Proof. First we prove the left inclusion in (12). If x∈[a,b] F (x) = ∅ then there is nothing to prove. Suppose , is a metric selection of F , since for any partition χ the function c χ,φ (x) ≡ p is a chain function corresponding to the chain φ = (p, . . . , p). Therefore, To show the right inclusion in (12), we use (11) and (9) and write In the case when k(x) ≥ 0 and b a k(x)dx = 0, the left inclusion in (13) follows directly from (12). To prove the right inclusion in (13), we start with (11). Denoting R = x∈[a,b] F (x) ∈ K(R d ) we get in view of the second property in (9) k(x)dx co(R), and the right inclusion follows.
Note that the middle set in (13)  6 The metric Fourier approximation of SVFs of bounded variation 6

.1 On Fourier approximation of real-valued functions of bounded variation
First we present the classical material relevant to our study of SVFs. For a 2π-periodic real-valued function f : R → R which is integrable over the period, its Fourier series is where a k = a k (f ) = 1 π For the partial sums of the Fourier series one has the well-known representation where ∂n,x(t) = Dn(x − t).
A basic result on the convergence of Fourier series of real-valued functions of bounded variation is the Dirichlet-Jordan Theorem (e.g., [39,Chapter II,(8

.1) Theorem]).
Dirichlet-Jordan Theorem. Let f : R → R be a 2π-periodic function of bounded variation on [−π, π]. Then at every point x In particular, Snf converges to f at every point of continuity of f . If f is continuous at every point of a closed interval I, then the convergence is uniform in I.
Following [39, Chapter II], we introduce the so-called modified Dirichlet kernel and the modified Fourier sum Clearly, We also need the next result that follows immediately from [39, Chapter II, (4.12) Theorem].
. Then its Fourier coefficients (14) satisfy the estimate

Extension to special classes of real-valued functions of bounded variation
It is known that functions of bounded variation with values in an arbitrary complete metric space (X, ρ) are not necessarily continuous, but have right and left limits at any point [15]. To study such functions we introduce the left and right local quasi-moduli for discontinuous functions of bounded variation.
and for x * ∈ [a, b) the right local quasi-modulus The facts given in the following remark are direct consequences of the above definitions.
Remark 6.5. Let f : [a, b] → X be a BV function and x * ∈ (a, b] for the left modulus or x * ∈ [a, b) for the right modulus, respectively.
(i) If f is monotone then (ii) Although at a point of discontinuity x * at least one of the local moduli ω − f, x * , δ , ω + f, x * , δ does not tend to zero as δ tends to zero, for the local quasi-moduli we always have An analogous relation holds for the right local quasi-modulus. Clearly, at a point of continuity of f the one sided local quasi-moduli and the one-sided local moduli of Section 3 coincide.
In the next two lemmas we derive results similar to those in Section 4 for the local one-sided moduli. Lemma 6.6. Let F ∈ F[a, b], x * ∈ (a, b] and c χ,φ be a chain function corresponding to a partition χ and a metric chain φ.
. By the definitions of the metric chain and of the chain function we have and we obtain the claim.
Proof. Let s ∈ S(F ) and δ > 0. There exists a sequence of chain functions {cn} n∈N that corresponds to a sequence of partitions {χn} n∈N with |χn| → 0 as n → ∞ such that s(x) = limn→∞ cn(x), x ∈ [a, b]. Take N ∈ N so large that |χn| < δ for all n ≥ N . We estimate − vs, x * , δ = vs(x * − 0) − vs(x * − δ). Take 0 < t < δ. For each n ≥ N we have by Lemma 6.6 By Theorem 3.8 we have Taking the limit as t → 0+ we obtain the claim.
Note that we cannot expect a bound for + vc χ,φ , x * , δ in terms of + vF , x * , δ + ε . The reason is that in the definition of the chain function we use values on the left of a point x * that we cannot control by + vF , x * , δ . However, the following estimates hold true for a metric selection s.
In the next definition we introduce several classes of periodic vector-valued functions.
Definition 6.9. Given B > 0, a point x ∈ R, a closed interval I ⊂ R and a modulus-bounding function ω, we define the following classes of functions.
for all 0 < δ ≤ π. Theorem 6.11. Let B > 0, x ∈ R and ω be a modulus-bounding function. Then for all f ∈ BV 1 B, x, ω and each δ ∈ (0, π] we have where C is the constant from Lemma 6.2. In view of Remark 6.3 one can take C = 2 in (20).
The next corollary follows from the above theorem.
For f ∈ BV 1 B, I, ω the estimate in the right-hand side of (20) does not depend on x ∈ I. We arrive at the following statement. Corollary 6.13. Let B > 0, I ⊂ R be a closed interval and ω be a modulus-bounding function satisfying lim δ→0+ ω(δ) = 0. Then Finally, if f is in addition continuous in I then the Fourier series of f converges to f on I, and the statement above takes the following form.

Extension to SVFs
We define the Fourier series of set-valued functions via the integral representation (15) using the weighted metric integral.
The metric Fourier series of F is the sequence of the set-valued functions {SnF } n∈N , where SnF is a SVF defined by ∂n,x(t)F (t)dt, x ∈ [−π, π], n ∈ N, whenever the integrals above exist.
For F ∈ F[−π, π] the integrals in Definition 6.15 exist. Moreover, each ∂n,x = Dn(x − ·) for fixed n ∈ N and x ∈ R is of bounded variation on each finite interval. Hence, if F ∈ F [−π, π], then the set-valued functions SnF have compact images by Proposition 5.10. By Theorem 5.7 we have Note that we do not expect metric selections s in this definition to be periodic. In fact, even if the set-valued function F itself is periodic, it can have metric selections that are not periodic (see Figure 6.16).
We show that this is the limit set of the Fourier approximants.
The next two theorems are the main results of the paper.
In case F is continuous, its Fourier series converges to F in the Hausdorff metric. Namely, the following holds true. Theorem 6.19. Let F ∈ F[−π, π] and let F be continuous at x ∈ (−π, π). Then lim n→∞ haus (S n F (x), F (x)) = 0.
If F is continuous in a closed interval I ⊂ (−π, π), then the convergence is uniform in I.
Proof. The first statement of the above theorem is an immediate consequence of Theorem 6.18. For the second statement note that there exists δ 0 > 0 such that [x − δ 0 , x + δ 0 ] ⊂ (−π, π) for all x ∈ I.
Defining ω(δ) as in the proof of Proposition 6.17 and applying Corollary 6.13, we obtain the result.

On the limit set of the Fourier approximants
In the previous section we proved that the sequence {S n F (x)} n∈N converges at a point x where F is discontinuous to the set A F (x) = 1 2 (s(x + 0) + s(x − 0)) : s ∈ S(F ) . An interesting question is to describe the set A F (x) in terms of the values of F . At the moment we do not have a satisfactory answer to this question.
The two statements below give some idea about the structure of a set-valued function F and its metric selections at a point x where F is discontinuous.
Proof. We show that F (x − 0) ⊆ F (x), the proof for F (x + 0) is similar.
Since F is bounded, we can restrict our consideration to a bounded region of R d , so that the convergence in the Hausdorff metric is equivalent to the convergence in the sense of Kuratowski (see Remark 2.2).
Consider y ∈ F (x − 0). Take an arbitrary sequence {x n } n∈N with x n < x, n ∈ N, and x n → x, n → ∞. Since F (x − 0) coincides with the lower Kuratowski limit lim inf t→x−0 F (t), for each n there exists y n ∈ F (x n ) such that y n → y, n → ∞. We have (x n , y n ) ∈ Graph(F ) for each n ∈ N and (x n , y n ) → (x, y), n → ∞. Since Graph(F ) is closed, it follows that (x, y) ∈ Graph(F ), and thus y ∈ F (x). This implies that F (x − 0) ⊆ F (x).
Proof. We prove the first claim, the proof of the second one is similar. Fix Clearly, F is left continuous at x and F ∈ F[a, b]. By Result 4.2 F has a representation by its metric selection. By Proposition 7.1 F (x) ⊆ F (x), and thus S( F ) ⊆ S(F ). Now, let y ∈ F (x − 0) = F (x) ⊆ F (x). There exists a selection s ∈ S( F ) ⊆ S(F ) such that y = s(x). Since F is left continuous at x, by Theorem 4.7 s is also left continuous at x. Thus, y = s(x) = s(x − 0) ∈ {s(x − 0) : s ∈ S(F )}.
In view of the last proposition and by the definition of A F (x) (see (22)), we conclude where the right-hand side is the Minkowski average which might be much larger than A F (x). One could conjecture that A F (x) coincides with the metric average of F (x−0) and F (x+0), namely It is easy to see that a sufficient condition for the inclusion is the property for any s ∈ S(F ). However, (27) is not always true. The next example provides a counterexample to both (27) and (26).
denote the closed disc of radius 1 with center at the point (x 1 , x 2 ), and let x ∈ (−π, π). Consider the function F : [−π, π] → K(R 2 ), F ∈ F[−π, π], defined by and its metric selection First we show that (27) does not hold. It is easy to see that s( 0)) is the projection of (0, 0) on F (x + 0). On the other hand, the pair s(x − 0), s(x + 0) is not a metric pair of (F (x − 0), F (x + 0)) since the line connecting the points s(x − 0) and s(x + 0) does not pass through any of the centers of the two discs. By similar geometric arguments one can show that , and that the selection s for which (27) does not hold satisfies s( Also the reverse inclusion to (26), , does not hold in general. The next example demonstrates this.
where x ∈ (−π, π). We have F (x − 0) = − 1 4 , 0, 1 4 , F (x + 0) = {−1, 1}, and their metric average is does not belong to A F (x), i.e., there is no metric selection s of F such that Indeed, if (28) is fulfilled for a selectionŝ of F , then for this selection we necessarily haveŝ(t) = 0 for t ∈ [−π, x) andŝ(t) = 1 + t − x for t ∈ (x, π] (with an arbitrary choice of the value s(x) ∈ F (x)). But suchŝ cannot be a metric selection, because there are no chain functions that would lead to such a selection. The only chain functions which might converge toŝ are constant with the value 0 on the left of x and piecewise constant functions with values sampled from 1 + t − x on the right of x, possibly except for the interval between two neighboring points of the partition that contains the point x. But no chain function can take the value 0 on the left of x and the value 1 + t − x on the right of x. Indeed, if x is not a point of the partition, then this is impossible because the closest point to 0 in the set F (t) = {−1 + t − x , 1 + t − x}, t > x, is −1 + t − x and not 1 + t − x, and the closest point to 1 + t − x in the set F (t) = − 1 4 , 0, 1 4 , t < x, is 1 4 and not 0. If x is a point of the partition, then the value of a chain function at x is one of the five values from F (x) = −1, − 1 4 , 0, 1 4 , 1 . The choices 0 and 1 are impossible because of the reasons explained above. But also the other three choices are impossible, since the pointwise limit of the chain functions would not be equal to 0 on the left of x.
Yet, the conjecture A F (x) = 1 2 F (x−0)⊕ 1 2 F (x+0) or a weaker form of it might be true for functions F from a certain subclass of F[a, b].