A constructive version of the extremum value theorem for spaces of vector–valued functions

It is shown that the extremum value theorem for spaces of two–dimensional vector–valued functions in an approximate format admits a proof in the sense of Bishop’s constructive mathematics. The proof is based on an explicit construction of functions that build an approximation to the original function space. 2010 Mathematics Subject Classification 03F60 (primary); 51M04 (secondary)


Introduction
The classical extremum value theorem is stated as follows: Theorem 1.1 Let f be a continuous function on a compact interval [a, b].Then, there exist real numbers x, y ∈ [a, b] such that f (x) = sup f and f (y) = inf f .This theorem can be generalized to compact metric spaces including function spaces.Consider, for example, the following optimization problem: where J is the cost functional, the variable t denotes time, x is called state and an element of some compact normed space X , also called state space, and function v is called control law.The function ϑ is called endpoint penalty and L is the Lagrangian, also called running cost.Such an optimization problem is, for example, addressed in optimal control.To solve (1) means to find a control law v * (x) such that J[v * ] = inf v J [v].For example, in case of robot navigation on a plane, we might look for the optimal ground speed vector as a function of the robot position.
Theorem 1.2 Let F be the space of all uniformly Lipschitz and uniformly bounded functions from a closed hypercube H(b, K) ⊂ R n to the plane R 2 , and let J be a uniformly continuous functional from F to R.Then, for any k ∈ N, there exists an The constructive notions used to prove this theorem are discussed in detail in Section 2.
Relaxing the statement of the theorem and considering approximate extrema can be justified for practical applications such as planar robot navigation.The key aspect of the proof of the main result is showing that, under certain conditions, the function space in question admits a finite approximation.

Preliminaries
For the proof of Theorem 1.2, we require certain notions from constructive mathematics and basic lemmas.For a comprehensive description, the reader may refer to Bishop and Bridges [3], Bridges and Richman [5], Bridges and Vita [6], Ye [13] and Schwichtenberg [10].First, we address the basics and recall briefly the notions of real numbers and sets in Section 2.1.After that, we discuss Euclidean spaces in Section 2.2.Finally, in Section 2.3, the constructive notions of functions and functionals are provided.

Basics
Bishop's constructive mathematics uses the notion of an operation which is an algorithm that produces a unique result in a finite number of steps for each input in its domain.
For example, a real number x is a regular Cauchy sequence of rational numbers in the sense that ∀n, n which means that a Cauchy sequence of rationals converges to the real it represents with 2n being the modulus of convergence.Convergence moduli for a sum and product of real numbers are easily defined.A Cauchy sequence of reals is also provided with a modulus of convergence.
A set is a pair of operations: ∈ determines that a given object is a member of the set, and = determines whenever two given set members are equal.Existence and universal quantifiers are interpreted as follows: ∃x ∈ A.ϕ[x] means that an operation has been derived that constructs an instance x along with a proof of x ∈ A, and a proof of the logical formula ϕ[x] as witnesses; ∀x ∈ A.ϕ[x] means that an operation has been derived that proves ϕ[x] for any x provided with a witness for x ∈ A. A set A is called inhabited if there exists an x ∈ A. A finite set is a set that admits a bijection to the set {1, 2, . . ., n} for some n ∈ N which means that all its elements are enumerable.

Euclidean space
The Euclidean space R n is a normed space with the norm x n i=1 (x i ) 2 1 2 where x i is the ith coordinate of x.The metric is defined as x − y for any x, y ∈ R n .We will use the notions of a ball and hypercube in which we focus solely on rational centers, radii and side lengths.A (rational A regular mesh on R n with a step δ ∈ Q is the set of points For example, a regular mesh on R with a step 1 is the set of points {. . ., −3, −2, −1, 0, 1, 2, 3, . . .} .

Pavel Osinenko and Stefan Streif
We will also consider regular meshes on closed balls and hypercubes.Due to Lemma 4.1 from Beeson [2, page 8], we can decide whether x = y or x = y for arbitrary algebraic numbers x, y.Notice that a closed ball is given by an expression that contains a polynomial of several variables and a radical.This allows deciding whether a given point belongs to the given closed ball (or hypercube) provided that the point possesses solely rational (or, in general, algebraic) coordinates.Consequently, distances between closed balls, hypercubes, their unions and boundaries, are algebraic numbers since their computation involves only finite products, sums and radicals of algebraic numbers.Further, we may define a regular mesh on a closed ball, respectively hypercube, as a finite set that is the intersection of the ball, respectively hypercube, with the regular mesh on the entire Euclidean space.The following lemma helps locate an arbitrary point in a closed hypercube: Lemma 2.1 Let P = {p i } i be a regular mesh on a closed hypercube H(0, K) ⊂ R n with a step δ ≤ K , and x ∈ H(0, K).Then, there exists a closed hypercube H( p i , δ), Proof Since x is a tuple of n real numbers (x 1 , . . ., x n ), we may assume that mth rational approximation to x is a tuple of rational numbers x(m) := (x 1 (n • m), . . ., Compute all distances ||x(m) − p i ||, p i ∈ P. Since they are algebraic numbers, we can decide whether ||x(m) − p i || < δ 2 .If this is the case for some p i , then x ∈ H( p i , δ).If there are more than one such balls, we pick the one with the smallest index.If ||x(m) − p j || = δ 2 for some indices j = j 1 , . . ., j L , then we pick the smallest such j and conclude that x ∈ B( p j , δ).
Remark Constructively, we cannot decide x ≤ 0 ∨ x ≥ 0 for an arbitrary real number x.Consequently, we cannot decide whether a point in a Euclidean space belongs to a given set.However, we can compare a real number with a non-trivial interval.In the lemma above, we generalize this result and use the fact that the hypercubes H( p i , δ) overlap.
Let B(0, K) ⊂ R n be a closed ball centered at the origin, and P = {p i } i be its mesh with a step δ .Let h denote the side length of closed hypercubes Hi ( p i , h) such that Here, ∂ denotes the boundary of a set.We call the algebraic number h δ := min {δ, h} the characteristic step of the mesh P = {p i } i .
Clearly, the union of the closed hypercubes Hi ( p i , h δ ) contains the closed ball B(0, K) and all closed hypercubes with adjacent centers have non-trivial intersections.The following shows that an arbitrary point can also be located inside a closed ball and is thus a direct consequence of Lemma 2.1: Lemma 2.2 Let P = {p i } i be a regular mesh on a closed ball B(0, K) ⊂ R n with a step δ .Let h δ denote its characteristic step.Suppose that x is an arbitrary point in B(0, K).
Then, there exists a closed ball

Functions and functionals
Let X denote a closed ball or hypercube in R n .A uniformly continuous (vector-valued) function from X to a R m is a pair consisting of an operation x → f (x), x ∈ X and an operation ω : Notice that there is no restriction in considering ω and ε as rational numbers since they are dense in the reals.If there exists a rational number L such that ∀x, y ∈ X. f (x) − f (y) ≤ L x − y , then the function f is Lipschitz continuous.We denote the normed space of all uniformly continuous vector-valued functions from X to R m by C(X, R m ).The corresponding norm is defined as f sup x∈X f (x) , which exists constructively since X is a compact set and every f is uniformly continuous.In the current work, we focus on subsets of C(X, R m ), m = 2 of all Lipschitz-continuous functions that have a common bound and a common Lipschitz constant.In justifying this assumption, we may claim that in a concrete application, the physical signals are limited in magnitude and rate of change.For example, consider the problem of optimal robot navigation on a plane.The vector of the robot's speed can be considered as a function of its position.The maximal magnitude of this vector is the maximal speed that the robot can achieve.The rate of change of the vector may be limited by the physical capabilities of the steering mechanisms.In order to evaluate separate control strategies, we require a performance mark.To this end, we use the notion of a uniformly continuous functional on some function space In this section, we prove Theorem 1.2.Let us recall its statement: Let F be the space of all uniformly Lipschitz and uniformly bounded functions from a closed hypercube X ⊂ R n to the plane R 2 , and let J be a uniformly continuous functional from F to R.Then, for any k ∈ N, there exists an f Proof The proof is organized into two parts.In the first, and the crucial, part, it is shown that the function space F is totally bounded.It is done by choosing a regular mesh on the target plane.It is then shown that any function from the said function space can be approximated by a piece-wise function which does not violate the Lipschitz constant.To this end, the Brehm's extension theorem is used [4].Provided that the points in question possess solely rational coordinates, its proof is within the Bishop's constructive mathematics.This is verified in the appendix.Finally, to achieve the required approximation precision, the step of the regular mesh needs to be chosen with special care.The second part is simpler and shows existence of the approximate extrema.The details are given below.

Part 1
Let L and K be the uniform Lipschitz constant and uniform bound for F respectively, and let X 0 = {x i } N i=1 be a finite sequence of unequal points in X .Without loss of generality, we may assume that all the points (x 1 , . . ., x N ) possess solely rational coordinates.We want to show that the subset is totally bounded in the following sense: for any k ∈ N, there exists a finite set {θ i } i of unequal elements of Θ such that for any θ ∈ Θ, there exists a θ j ∈ {θ i } such that θ − θ j Θ ≤ 1 k .To this end, let P = {p i } i be a regular mesh on the closed ball B(0, K) ⊂ R 2 with a step δ and characteristic step h δ .Let f be an arbitrary function in the space F .Fix an arbitrary l ∈ N. We construct an approximating function ψ : lN .First, the image of ψ on X 0 is constructed inductively.By Lemma 2.2, there exists a closed ball B( p j 1 , √ 2h δ ), p j 1 ∈ P that contains f (x 1 ).We set ψ(x 1 ) := p j 1 .Assume that f (x 2 ) ∈ B( p j 2 , r 1 ), p j 2 ∈ P, r 1 := √ 2h δ .We pick a point p j 2 closest to p j 2 such that which, by setting ψ(x 1 ) := p j 2 , satisfies the Lipschitz condition for ψ .At this step, we may take a closed ball B( p j 2 , r 2 ) that contains B( p j 2 , r 1 ), and in turn f (x 2 ), with a radius r 2 that is larger than r 1 by a value that depends only on the choice of the initial mesh step δ .Since the total number of steps is fixed by N , setting δ sufficiently small ensures that all ψ(X 0 ) are within 1  lN from the respective f (X 0 ) while preserving the Lipschitz constant.Now, we need to extend ψ to the whole X .First, we may assume that L = 1 without loss of generality since, otherwise, we can scale the X accordingly.If n > 2, we project the points of X onto R 2 .Denote this projection by π 2 .Notice that π 2 has a Lipschitz constant equal to one and, clearly, it projects points with rational coordinates onto points with rational coordinates.Then, we apply Theorem 4.1 from Section 4 to extend π 2 • ψ onto the entire X .Precomposing the resulting function with the projection onto the ball B(0, K) ensures that it belongs to the space F .If n ≤ 2, the extension theorem applies directly and no projection onto the two-dimensional subspace is required.Let us denote the resulting function by ψ .
So far, we showed how to construct an arbitrarily close approximation to f ∈ F .Since each such an approximating function is uniquely defined by its values at finitely many points X 0 , and since the distances between the approximating function values at each two points of X 0 have fixed bounds, there are finitely many such functions.Further, since f was arbitrary, it follows that Θ is totally bounded.By the constructive Arzela-Ascoli's lemma (Bishop and Bridges [3, page 100]), the function space F is totally bounded as well.Next, we find an approximate extremum of J .

Part 2
Since F is totally bounded and J is uniformly continuous, inf J exists.Let α be the continuity modulus of J and F 0 = {f 1 , . . ., f N } be an α( 1 8k )-approximation to F .Consider all finitely many {J[ f i ](8k)} , i = 1, . . ., N .Let J[ f j ](8k) be the smallest one, and such that j is the smallest index if there are more than one such indices.Observe that |J[ k is a lower bound of J and so, in particular, inf J ≥ J[ f j ] − 1 k .

Conclusion
In the present work, we addressed the extremum value theorem for spaces of twodimensional vector-valued functions on Euclidean spaces from the standpoint of constructive mathematics.By showing that spaces of uniformly Lipschitz and uniformly bounded functions from a closed hypercube to a plane are totally bounded, an approximate extremum of the respective uniformly continuous functional can be constructed.The theorem proven may be used for purposes of formal verification in certain applications such as optimal control.

Appendix
We call a point in a Euclidean space R n algebraic if its coordinates are solely algebraic numbers.
A (closed) polyhedron is an inhabited set of points of the Euclidean space satisfying linear inequalities Ax ≤ b, A ∈ R n×n , b ∈ R n×1 .If the entries of A and b are solely algebraic numbers, then the polyhedron is called algebraic.If a polytope is a union of solely algebraic polyhedrons, then it is algebraic as well.A polyhedron (respectively, polytope) P is bounded if there exists a rational number x such that x ≤ x for any x in P.An n-dimensional simplex is a convex hull of n + 1 affinely independent points.A triangulation of a bounded algebraic n-dimensional polyhedron P is a finite set {T i } i of algebraic simplices whose intersections are at most (n − 1)-dimensional and such that P = ∪ i T i .For example, a triangulation of a two-dimensional polyhedron is a collection of non-degenerate triangles that may have a common vertex or segment of an edge, but no two-dimensional intersection.
A motion in the plane R 2 is a function f that is a composition of a translation, rotation and reflection.Clearly, it is distance-preserving in the sense that f (x)−f (y) = x−y for any x, y.Translation, rotation and reflection can be described as linear transformations of the form x → Tx where T is the transformation matrix.A motion can be thus described by a transformation matrix as well.Notice that a motion is always invertible since the corresponding transformation matrix is regular.A motion is called algebraic if the corresponding transformation matrix comprises solely of algebraic entries.For example, A constructive version of the extremum value theorem for spaces of vector-valued functions 9 is an algebraic motion and describes the clockwise rotation by the angle arcsin 3  5 .In contrast, f (x) = x + π is not an algebraic motion.Any three algebraic points forming a non-degenerate triangle can be moved by an algebraic motion to new algebraic points preserving the respective distances (Petrunin and Yashinski [8]).An algebraic piecewise motion f on a bounded algebraic two-dimensional polyhedron P is a pair of a triangulation {T i } i of the polyhedron and a collection of algebraic motions {f i } i such that f | Ti = f i .For example, folding of a piece of paper without ripping can be considered as an algebraic piecewise motion if foldings are performed at algebraic points.Notice that each algebraic piecewise motion has a triangulation and a collection of algebraic motions as witnesses.Clearly, an algebraic piecewise motion is non-expansive in the sense that f (x) − f (y) ≤ x − y for any x, y.The notion of an algebraic piecewise motion can be directly generalized to algebraic polytopes.Constructively, we do not have the full power of set operations in a Euclidean space.For example, if A and B are arbitrary sets in R n , we cannot decide whether A ∩ B = ∅ or A∩B is inhabited.In the current work, we are not concerned with arbitrary set operations and limit ourselves to the class of semi-algebraic sets of the form x : where "•" denotes "<","≤" or "=" and f ij (x) and g ij (x) are algebraic piecewise motions on algebraic polytopes.Let us denote this class by AS .For example, an algebraic polytope itself belongs to AS .Further, the set complement of a set A ∈ AS , denoted by R n \A, is, again, a set of the class AS (it can be done by transforming the sign "<","≤" or "=" in the respective formula).Notice that if f ij (x) and g ij (x) are algebraic motions, each ≥ 0 which is a collection of algebraic inequalities.The same applies if f ij (x) and g ij (x) are algebraic piecewise motions by considering the inequalities on the simplices where f ij (x) and g ij (x) are both algebraic motions.Working solely with sets of the class AS allows us to perform the ordinary set operations via solving the respective systems of algebraic equations and inequalities.A similar strategy of working with semi-algebraic sets was already used in formal verification of control systems by Platzer [9] based on the quantifier elimination on real closed fields Tarski [11].Working in AS will be used in Section 4 to verify that certain vector-valued functions on finite subsets of R 2 admit non-expansive extensions.It is shown that the Brehm's extension theorem is valid in the Bishop's constructive mathematics provided that the points in question are rational.

Extension theorem
For the proof of Theorem 1.2, we require the following result: , {b i } n i=1 be finite subsets of points in R 2 with rational coordinates such that ∀i, j. b i − b j ≤ a i − a j .Let A be the convex hull of {a i } n i=1 and X be a closed hypercube containing A. Then, there exists a non-expansive function f : This theorem was first addressed by Kirszbraun [7] and Valentine [12], and then revisited by Brehm [4].A similar proof can be found in Akopyan and Tarasov [1] and Petrunin and Yashinski [8, page 21].In our restricted case, we consider only points with rational coordinates.In the following, we verify that the proof is constructive using the notions introduced in Section 2.2.
Proof We first construct an algebraic distance-preserving function h on A with the required properties by induction on the number of the points.If n = 1, we may take h(x) := x + (b 1 − a 1 ) which is clearly an algebraic motion on the entire space.Suppose that an algebraic piecewise motion g : A → R 2 , such that ∀i = 1, . . ., n − 1.g(a i ) = b i , was constructed.Define a set Ω := {x : x ∈ A ∧ a n − x < b n − g(x) }.Since g is algebraic, we can decide whether b n = g(a n ) or b n = g(a n ).In the former case, we take h to be g.In the latter, Ω is inhabited since a n ∈ Ω.We show that if x belongs to Ω, then so does the line segment between a n and x.Take a point y in this line segment.Then The function g is an algebraic piecewise motion which implies where the last line follows from the triangle inequality g(x) − g(y) ≤ b n − g(y) + b n − g(y) .Since a n − y < b n − g(y) , it follows that y ∈ Ω.We now inspect how the boundary ∂ A Ω := ∂Ω ∩ A is constructed.Let {T i } i be the triangulation of A such that g on each triangle is a motion g i .Let c n := g −1 i (a n ).Notice that c n is an algebraic point.Since g i is a motion and g| Ti = g i , for any x ∈ T i , we have A constructive version of the extremum value theorem for spaces of vector-valued functions11 Since Ω and T i belong to AS , we can decide whether the intersection Ω ∩ T i is inhabited.Suppose it is inhabited.Consider the line We have Matching (4) with ( 5) using (3), we see that ∂ A Ω ∩ T i is a line segment.Since {T i } i is a finite set, ∂ A Ω is a finite collection of line segments.Consider a line segment ω i of ∂ A Ω. Let τ i be the triangle formed by a n and ω i .Let h i be an algebraic motion that maps a n to b n and the endpoints of ω i to their respective positions under g i .For x ∈ ω i , we have g(x) = g i (x) and so g(x) = h i (x).Let h| τi := h i and h| A\Ω := g.Further, since ∂Ω, ∂ A Ω ∈ AS , we can decide whether ∆ := ∂Ω ∩ ∂ A Ω is inhabited.If this is the case ( for otherwise, we are done), consider the algebraic polytopes D k , k = 1, . . ., m formed by the endpoints of ∆ that lie on ∂Ω ∩ A and the line segments from these endpoints to a n .Let λ 1 and λ 2 denote the said endpoints for some algebraic polytope D k .Since h coincides with g on the line segments [a n , λ 1 ] and [a n , λ 2 ], and, moreover, it acts as algebraic motions on these line segments, and since g is non-expansive, it follows that: ) to g(λ 2 ) which is the chord of the circle on which the point λ 2 slid to the new position.Take the middle point of the chord and fold D k around the ray going from a n to this middle point so that λ 2 matches with g(λ 2 ).The resulting function is thus constructed by translating and rotating the whole D k and then reflecting the fragment-to-fold around the said ray which constitutes a piecewise motion.This motion is clearly algebraic since all the points involved are algebraic.So far, we constructed an algebraic piecewise motion h : A → R 2 such that ∀i.h(a i ) = b i .
To construct a required function f on the whole X , observe that the projection onto an algebraic polyhedron is non-expansive.In showing this, we use the fact that an algebraic polyhedron is convex: suppose that x and y are points and r and s are their has the derivative with respect to t at zero being equal to 2 r − s, x − r − y + s which is non-negative due to the above two inequalities.It follows that γ is an increasing function and, in particular, γ(1) ≥ γ(0) whence x − y ≥ r − s .Therefore, we complete the construction of f by projecting the points of X onto A.

λ 1 −
a n = g(λ 1 ) − b n λ 2 − a n = g(λ 2 ) − b n g(λ 1 ) − g(λ 2 ) ≤ λ 1 − λ 2We can construct the required function on D k as follows.Translate and rotate D k so that [a n , λ 1 ] coincides with [b n , g(λ 1 )].This can be done since the initial and the new vertices of D k are algebraic.So far, the line segment [a n , λ 2 ] "turned" around b n closer to [a n , λ 1 ].Draw a line segment from g(λ 1