aa r X i v : . [ m a t h . M G ] O c t A note on Brehm’s extension theorem
P. Osinenko a a Laboratory for Automatic Control and System Dynamics; Technische Universität Chemnitz, 09107 Chemnitz, Germany
Abstract
Brehm’s extension theorem states that a non–expansive map on a finite subset of a Euclidean space canbe extended to a piecewise–linear map on the entire space. In this note, it is verified that the proof ofthe theorem is constructive provided that the finite subset consists of points with rational coordinates.Additionally, the initial non–expansive map needs to send points with rational coordinates to points withrational coordinates. The two–dimensional case is considered.
Keywords: constructive mathematics, extension, Euclidean space
1. Introduction
Brehm’s extension theorem is stated as follows:
Theorem 1.
Let M be a finite subset of R n , and ϕ : M → R m , m ≤ n a map with the property thatfor any a, b ∈ M , the condition k ϕ ( a ) − ϕ ( b ) k ≤ k a − b k holds. Then, there exists a piecewise–linear map f : R n → R m such that ∀ a ∈ M.f ( a ) = ϕ ( a ) . Here, k•k denotes the Euclidean distance. A map with the property described is also called non–expansive .The theorem was first addressed by Kirszbraun (1934) and Valentine (1943), and then revisited by Brehm(1981). A similar proof can be found in Akopyan and Tarasov (2008) and Petrunin and Yashinski (2014,p. 21). In the present work, it is verified that the theorem admits a constructive proof in the sense ofBishop’s constructive mathematics (Bishop and Bridges, 1985) provided that M and ϕ ( M ) consist of pointswith rational coordinates. Only the planar case m = n = 2 is considered.
2. Preliminaries
In this section, selected basics of constructive mathematics are briefly discussed. For a comprehensivedescription, refer, for example, to (Bishop and Bridges, 1985; Bridges and Richman, 1987; Bridges and Vita,2007; Ye, 2011; Schwichtenberg, 2012). 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, m ∈ N . | x ( n ) − x ( m ) | ≤ n + 1 m where x ( n ) is an operation that produces the n th rational approximation to x . 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 areequal. Existence and universal quantifiers are interpreted as follows: ∃ x ∈ A.ϕ [ x ] means that an operationhas 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 providedwith a witness for x ∈ A . A set A is called inhabited if there exists an x ∈ A . A finite set is a set that Email address: [email protected] (P. Osinenko) dmits a bijection to a set { , , ...n } for some n ∈ N which means that all its elements are enumerable. TheEuclidean space R n is a normed space with the norm k x k , (cid:0)P ni =1 x i (cid:1) / where x i is the i –th coordinate of x . The metric is defined as k x − y k for any x, y ∈ R n . A point x in the Euclidean space is called algebraic if its coordinates are algebraic numbers.A (closed) polytope is a union of polyhedrons whereas a (closed) polyhedron is an inhabited set of pointsof the Euclidean space satisfying linear inequalities Ax ≤ b, A ∈ R n × n , b ∈ R n × . If the entries of A and b aresolely algebraic numbers, then the polyhedron is called algebraic . If a polytope is a union of solely algebraicpolyhedrons, then it is algebraic as well. A polyhedron (respectively, polytope) P is bounded if there existsa rational number ¯ x such that k x k ≤ ¯ 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 isa finite set { T i } i of algebraic simplices whose intersections are at most ( n − –dimensional and such that P = ∪ i T i . For example, a triangulation of a two–dimensional polyhedron is a collection of non–degeneratetriangles that may have a common vertex or segment of an edge, but no two–dimensional intersection.A motion in the plane R is a map f that is a composition of a translate, a rotation and a reflection.Clearly, it is distance–preserving in the sense that k f ( x ) − f ( y ) k = k x − y k for any x, y . Translate, rotationand reflection can be described as linear transformations of the form x T x where T is the transformationmatrix. A motion can be thus described by a transformation matrix as well. Notice that a motion isalways invertible since the corresponding transformation matrix is regular. A motion is called algebraic ifthe corresponding transformation matrix comprises solely of algebraic entries. For example, f ( x ) = q −
925 35 − q − x is an algebraic motion and describes the clockwise rotation by the angle arcsin . In contrast, f ( x ) = x + π isnot an algebraic motion. Any three algebraic points forming a non–degenerate triangle can be moved by analgebraic motion to new algebraic points preserving the respective distances (Petrunin and Yashinski, 2014).An algebraic piecewise–linear map f on a bounded algebraic two–dimensional polyhedron P is a pairof a triangulation { T i } i of the polyhedron and a collection of algebraic motions { f i } i such that f | T i = f i .For example, folding of a piece of paper without ripping can be considered as an algebraic piecewise–linearmap if foldings are performed at algebraic points. Notice that each algebraic piecewise–linear map has atriangulation and a collection of algebraic motions as witnesses. Clearly, an algebraic piecewise–linear mapis non–expansive. The notion of an algebraic piecewise–linear map can be directly generalized to algebraicpolytopes.It is important to notice that, for arbitrary real numbers x, y , it is not decidable whether x = y or x = y .This limitation has a number of consequences for the theory of the Euclidean space R n . In particular, nofull power of set operations is available. For example, if A and B are arbitrary sets in R n , it is not decidablewhether A ∩ B = ∅ or A ∩ B is inhabited. In this note, set operations are limited to the class of sets of theform n x : W Ni =1 V M i j =1 E ij o with E ij being a formula of the type A ij x • b ij or k f ij ( x ) k • k g ij ( x ) k where “ • ”denotes “ < ”,” ≤ ” or ” = ” and f ij and g ij are algebraic piecewise–linear maps on algebraic polytopes. Denotethis class by AS . For example, an algebraic polytope itself belongs to AS . Further, the set complementof a set A ∈ AS , denoted by R n \ A is, again, an element of the class AS (it can be done by transformingthe sign “ < ”,” ≤ ” or ” = ” in the respective formula). Notice that if f ij and g ij are algebraic motions, each k f ij ( x ) k ≤ k g ij ( x ) k is equivalent to P nk =1 ( f ij ( x )) k ≤ P nk =1 ( g ij ( x )) k . The same applies if f ij and g ij arealgebraic piecewise–linear maps by considering the inequalities on the simplices where f ij and g ij are bothalgebraic motions. If a set A has the form n x : V Ni =1 k f i ( x ) k < k g i ( x ) k o with f i and g i algebraic piecewise–linear, then its boundary is defined to be the set ∂A , n x : V Ni =1 k f i ( x ) k = k g i ( x ) k o which in turn belongsto AS . Lemma 4.1 from Beeson (1980, p. 8) states decidability of equality over the field of algebraic realnumbers. This allows performing the ordinary set operations on the sets of the described class. In thefollowing, the extension theorem is revisited and verified to admit a constructive proof.2 . Extension theorem The proof of the following theorem is mostly based on (Akopyan and Tarasov, 2008; Petrunin and Yashinski,2014).
Theorem 2.
Let { a i } ni =1 , { b i } ni =1 be finite subsets of points in R with rational coordinates such that ∀ i, j. k b i − b j k ≤ k a i − a j k . Let A be the convex hull of { a i } ni =1 . Then, there exists an algebraic piecewise–linearmap f : A → R such that ∀ i.f ( a i ) = b i .Proof. The theorem is proven by induction on the number of points. If n = 1 , one may take f ( x ) := x + ( b − a ) which is clearly an algebraic motion on the entire space. Suppose that an algebraic piecewise–linear map g : A → R , such that ∀ i = 1 , . . . , n − .g ( a i ) = b i , was constructed. Define a set Ω := { x : x ∈ A ∧ k a n − x k < k b n − g ( x ) k} . Since g is algebraic, it is decidable whether b n = g ( a n ) or b n = g ( a n ) .In the former case, take f to be g . In the latter, Ω is inhabited since a n ∈ Ω . Notice 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, k a n − y k + k y − x k = k a n − x k . Since x ∈ Ω , k a n − x k ≤ k b n − g ( x ) k . The map g is an algebraic piecewise–linear which implies k g ( x ) − g ( y ) k ≤ k x − y k . Therefore, k a n − y k = k a n − x k − k y − x k < k b n − g ( x ) k − k g ( x ) − g ( y ) k≤ k b n − g ( y ) k (1)where the last line follows from the triangle inequality k g ( x ) − g ( y ) k ≤ k b n − g ( y ) k + k b n − g ( y ) k . Since k a n − y k < k b n − g ( y ) k , it follows that y ∈ Ω . Now, the boundary ∂ A Ω := ∂ Ω ∩ A is inspected. Let { T i } i bethe triangulation of A such that g on each triangle is a motion g i . Let c n := g − i ( a n ) . Notice that c n is analgebraic point. Since g i is a motion and g | T i = g i , for any x ∈ T i , it follows that k c n − x k = k b n − g ( x ) k . (2)Since Ω and T i belong to AS , it is decidable whether the intersection Ω ∩ T i is inhabited. Suppose it isinhabited. Then, consider the line l i := { x : k x − a n k = k x − c n k} . It follows that: ∂ A Ω ∩ T i = { x : k a n − x k = k b n − g ( x ) k ∧ x ∈ T i ∧ x ∈ A } (3)and l i ∩ T i ∩ A = { x : k x − a n k = k x − c n k ∧ x ∈ T i ∧ x ∈ A } . (4)Matching (3) with (4) using (2), one can 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 f i be an algebraic motion that maps a n to b n and the endpoints of ω i to their respective positionsunder g i . For x ∈ ω i , it follows that g ( x ) = g i ( x ) and so g ( x ) = f i ( x ) . Let f | τ i := f i and f | A \ Ω := g . Further,since ∂ Ω , ∂ A Ω ∈ AS , it is decidable whether ∆ := ∂ Ω ∩ ∂ A Ω is inhabited. If this is the case (for otherwise,the result is trivial), consider the algebraic polytopes D k , k = 1 , . . . , m formed by the endpoints of ∆ thatlie on ∂ Ω ∩ A and the line segments from these endpoints to a n . Let λ and λ denote the said endpoints3or some algebraic polytope D k . Since f coincides with g on the line segments [ a n , λ ] and [ a n , λ ] , and,moreover, it acts as algebraic motions on these line segments, and since g is non–expansive, it follows that k λ − a n k = k g ( λ ) − b n k , k λ − a n k = k g ( λ ) − b n kk g ( λ ) − g ( λ ) k ≤k λ − λ k . The required map on D k can be constructed as follows. Translate and rotate D k so that [ a n , λ ] coincideswith [ b n , g ( λ )] . This can be done since the initial and the new vertices of D k are algebraic. So far, the linesegment [ a n , λ ] “turned” around b n closer to [ a n , λ ] . Draw a line segment from g ( λ ) to g ( λ ) which is thechord of the circle on which the point λ slid to the new position. Take the middle point of the chord andfold D k around the ray going from a n to this middle point so that λ matches with g ( λ ) . The resultingmap is thus constructed by translating and rotating the whole D k and then reflecting the fragment—to–foldaround the said ray which constitutes a piecewise–linear map. This map is clearly algebraic since all thepoints involved are algebraic. References
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