Simple close curve magnetization and application to Bellman's lost in the forest problem
aa r X i v : . [ m a t h . G M ] J a n SIMPLE CLOSE CURVE MAGNETIZATION AND APPLICATIONTO BELLMAN’S LOST IN THE FOREST PROBLEM
T. AGAMA
Abstract.
In this paper we introduce and develop the notion of simple closecurve magnetization. We provide an application to Bellman’s lost in the for-est problem assuming special geometric conditions between the hiker and theboundary of the forest. Introduction and problem statement
Bellman’s lost in the forest problem is a central problem which lies at the interfaceof geometry and optimization. It has its origin dating back in 1995 by the Americanapplied mathematician
Richard Bellman [1]. The problem has the following well-known formulation which can be found in [2]
Question 1 (Bellman’s lost in the forest problem) . Given a hiker in the forest withhis orientation within the forest unbeknownst to him, then what is the best possibledecision to be taken to exit in the shortest possible time taking into considerationthe shape of the forest and the dimension of the space covering the forest?Much work has been done in studying this problem by a few authors (see [3]). Thispresumably has not to do with diminished interest and less popularity but maypartly be attributed to the conceivable complexity of any viable tool to studyingthe problem. Regardless of its inherent difficulty a complete solution has beenfound and is known for a few class of shapes which are taken to be forest [3].In this paper we introduce and develop the notion of magnetization of simple closecurves. Using this notion we devise an algorithm that takes as input the shape ofthe forest and the dimension within which the forest resides and produce as outputthe optimal path to be taken by the hiker ~v ∈ R n .2. Simple close curves with magnetic boundaries
In this section we introduce the notion of simple close curve with magnetic bound-aries equipped magnets. We study this concept and expose some connections withother notions. . Date : January 5, 2021.2000
Mathematics Subject Classification.
Primary 54C40, 14E20; Secondary 46E25, 20C20.
Key words and phrases. simple close curve; magnetization; magnetic field; Bellman; forest;hiker.
Definition 2.1.
Let C be a simple close curve in R n and V n be the space of vectorsin R n . Then by the magnetization of the interior of C , denoted C Int , with magnets ~u , . . . ~u k on the boundary C B , we mean the mapΛ ~u ,~u ,...,~u k : C Int −→ V n such that for any ~v ∈ C Int with ~v = ~O thenΛ ~u ,~u ,...,~u k ( ~v ) = Λ ~u j ( ~v ) = ~u j − ~v with ~v · ~u j = 0 if and only if || ~v − ~u j || = min {|| ~u s − ~v ||} ks =1 . We call || Λ ~u ,~u ,...,~u k ( ~v ) || = || Λ ~u j ( ~v ) || = || ~v − ~u j || the measure of magnetization. We denote the magnetic field of the magnet ~u j by O ~u j . Strictly speaking, we denote the ǫ -magnetic field of the magnet ~u j by O ~u j ( ǫ )and we say ~v i ∈ O ~u j ( ǫ ) if and only || ~u j − ~v i || ≤ ǫ. Definition 2.2.
Let C be a simple close curve in R n with boundary C B . Then wesay the boundary is magnetic with magnets ∪ ∞ i =1 { ~u i } ∈ C B if for any ~u j ∈ C B thereexist some ~u i ∈ ∪ ∞ i =1 { ~u i } with ~u i ∈ O ~u j ( ǫ )for any ǫ > O ~u i ( δ ) \ C Int = ∅ and O ~u i ( δ ) \ ( R n \ C B ∪ C Int ) = ∅ for any δ >
0. We denote with B ( O ~u j ( ǫ ))the boundary of the neighbourhood O ~u j ( ǫ ).The language as espoused in Definition 2.2 suggests very clearly that magnets areallowed to be dense on the boundary of their simple close C with magnetic boundary C B . This strict enforcement will ease the development of our geometry and relatedtheories. In any case we can take the infinite set of magnets on the boundary of atypical simple close curve to be the entire boundary C B ⊂ R n . Remark . Next we put all simple close curves with magnetic boundaries equippedwith magnets and their constant dilates into one single category. In essence wewould consider two simple close curves with magnetic boundaries to be distinct ifthere exists a magnet on the boundary of one curve which fails to be a constantdilate on the other.
IMPLE CLOSE CURVE MAGNETIZATION AND APPLICATION TO BELLMAN’S LOST IN THE FOREST PROBLEM3
Proposition 2.1.
Every simple close curve C in R n with magnetic boundary isuniquely determined by their magnetic boundary C B upto constant dilates of theirmagnets.Proof. Let C be a simple closed curve with magnetic boundary C B in R n endowedwith magnets ∪ ∞ i =1 { ~s i } and ∪ ∞ i =1 { ~t i } . Pick arbitrarily ~v ∈ C Int then applying themagnetization Λ : C Int −→ V n on the interior C Int , we have the equalityΛ ∪ ∞ i =1 { ~s i } ( ~v ) = Λ ~s j ( ~v )for some 1 ≤ j with ~v · ~s j = 0 if and only if || ~v − ~s j || = min {|| ~s i − ~v ||} ∞ i =1 andΛ ∪ ∞ i =1 { ~t i } ( ~v ) = Λ ~t k ( ~v )for some 1 ≤ k with ~t k · ~v = 0 if and only if || ~v − ~t k || = min (cid:8) || ~t i − ~v || (cid:9) ∞ i =1 . It followsthat λ~s j = ~t k for λ ∈ R . Since the vector ~v is an arbitrary point in C Int the claimfollows immediately. (cid:3)
Theorem 2.4.
Let C be a simple closed curve in R n with magnetic boundary C B equipped with magnets ∪ ∞ i =1 { ~u i } . For any ~v ∈ C Int such that ~v = ~O then Λ ∪ ∞ i =1 { ~u i } ( ~v ) = Λ ~u j ( ~v ) with ~v · ~u j = 0 if and only if there exists ǫ > such that ~u j ∈ O ~v ( ǫ ) and ~u s
6∈ O ~v ( ǫ ) for all s = j .Proof. First let C be a simple closed curve in R n with magnetic boundary C B equipped with magnets ∪ ∞ i =1 { ~u i } . Let us pick arbitrarily ~v ∈ C Int such that ~v = ~O ,apply the magnetization Λ : C Int −→ V n and supposeΛ ∪ ∞ i =1 { ~u i } ( ~v ) = Λ ~u j ( ~v )with ~v · ~u j = 0, then || ~v − ~u j || = min {|| ~u s − ~v ||} ∞ s =1 for all ~u s ∈ ∪ ∞ i =1 { ~u i } . Nowchoose ǫ = || ~u j − ~v || and construct the neighbourhood O ~v ( ǫ ). By virtue of theconstruction ~u j ∈ O ~v ( ǫ ). We claim that ~u s
6∈ O ~v ( ǫ ) for all s = j with ~u s ∈ ∪ ∞ i =1 { ~u i } . Let us suppose to the contrary that there exist at least some ~u t ∈ ∪ ∞ i =1 { ~u i } ⊂ C B with t = j such that ~u t ∈ O ~v ( ǫ ). Then there exist some ~u k ∈ ∪ ∞ i =1 { ~u i } ⊂ C B with k = j such that ~u k ∈ B ( O ~v ( ǫ )) ∩ C B so that || ~v − ~u k || = ǫ = min {|| ~u s − ~v ||} ∞ s =1 . Applying the magnetization, It follows thatΛ ∪ ∞ i =1 { ~u i } ( ~v ) = Λ ~u k ( ~v )with ~v · ~u k = 0. The upshot is that ~u j = λ~u k for λ = 1 so that ǫ = || ~v − ~u k || 6 = || ~v − ~u j || T. AGAMA which contradicts our choice of || ~v − ~u j || = ǫ > ǫ > ~u j ∈ O ~v ( ǫ ) and ~u s
6∈ O ~v ( ǫ )for all s = j . Then || ~v − ~u j || = min {|| ~u s − ~v ||} ∞ s =1 for all ~u s ∈ ∪ ∞ i =1 { ~u i } and ~v ∈ C Int . The claim follows immediately from thisassertion by applying the magnetization Λ : C Int −→ V n . (cid:3) Theorem 2.5.
Let C be a simple close curve in R n with magnetic boundary C B equipped with magnets ∪ ∞ i =1 { ~u i } . Then for any ~v ∈ C Int such that ~v = ~O thereexists some ǫ > such that ~u j ∈ O ~v ( ǫ ) for some ~u j ∈ ∪ ∞ i =1 { ~u i } and ~u t
6∈ O ~v ( ǫ ) with t = j for all ~u t ∈ ∪ ∞ i =1 { ~u i } .Proof. Let C B and C Int be the magnetic boundary and the interior of the simpleclose curve C , respectively. Let ∪ ∞ i =1 { ~u i } ⊂ C B be the magnets on the boundary.Pick arbitrarily ~v ∈ C Int such that ~v = ~O , then for each ~u j ∈ ∪ ∞ i =1 { ~u i } there exists ǫ > ~u j ∈ O ~v ( ǫ ) . Let us choose O ~v ( δ ) = min {O ~v ( ǫ ) | ~u j ∈ O ~v ( ǫ ) , ǫ > } (2.1)for each ~u j ∈ ∪ ∞ i =1 { ~u i } . Let ~u j ∈ O ~v ( δ ) then it follows that || ~u j − ~v || = δ by virtueof (4.1) and || ~u j − ~v || = δ = min {|| ~v − ~u s || : ~u s ∈ ∪ ∞ i =1 { ~u i }} . (2.2)Apply the magnetization Λ : C Int −→ V n , then we haveΛ ∪ ∞ i =1 { ~u i } ( ~v ) = Λ ~u j ( ~v )with ~u j · ~v = 0. Let us suppose to the contrary ~u s ∈ O ~v ( δ ) for at least some ~u s ∈ ∪ ∞ i =1 { ~u i } with j = s . Then it follows that there exists some ~u t ∈ ∪ ∞ i =1 { ~u i } with t = j such that ~u t ∈ C B ∩ B ( O ~v ( δ )) . It follows that || ~v − ~u t || = δ = min {|| ~v − ~u s || : ~u s ∈ ∪ ∞ i =1 { ~u i }} so that by applying the magnetization Λ : C Int −→ V n , we haveΛ ∪ ∞ i =1 { ~u i } ( ~v ) = Λ ~u t ( ~v )with ~u t · ~v = 0. It follows that there exists some α = 1 such that ~u t = α~u j so that || ~v − ~u t || = δ = || ~v − ~u j || thereby contradicting (2.2). This completes the proof of the theorem. (cid:3) IMPLE CLOSE CURVE MAGNETIZATION AND APPLICATION TO BELLMAN’S LOST IN THE FOREST PROBLEM5 Connected and isomorphic simple close curves with magneticboundaries
In this section we introduce a classification scheme for all simple close curves C in R n with magnetic boundaries C B . This scheme pretty much allows us to put allsimilar such simple close curves into a single family and choose a representative forour work. Definition 3.1.
Let C and C be simple close curves with magnetic boundariesequipped with magnets ∪ ∞ i =1 { ~t i } and ∪ ∞ i =1 { ~w i } , respectively. Then we say C and C are connected if there exists some ~t j ∈ ∪ ∞ i =1 { ~t i } and some ~w k ∈ ∪ ∞ i =1 { ~w i } suchthat ~t j = λ ~w k for some λ ∈ R . We denote the connection by C ⇌ C . We say C and C are isomorphic if the connection exists for each ~t j ∈ ∪ ∞ i =1 { ~t i } and ~w k ∈ ∪ ∞ i =1 { ~w i } .We denote the isomorphism by C ≍ C . Proposition 3.1.
Let C and C be simple close curves with magnetic boundariesequipped with magnets ∪ ∞ i =1 { ~t i } and ∪ ∞ i =1 { ~w i } , respectively. If C ⊂ C then C ≍ C .Proof. Suppose C ⊂ C and let C Int and C Int be their interior, respectively. Nextpick arbitrarily a point ~v ∈ C Int such that ~v = ~O , then ~v ∈ C Int so thatmin (cid:8) || ~t s − ~v || (cid:9) ∞ s =1 ≤ min {|| ~w s − ~v ||} ∞ s =1 . Let us set || ~t j − ~v || = min (cid:8) || ~t s − ~v || : ~t s ∈ ∪ ∞ i =1 { ~t i } (cid:9) ∞ s =1 and || ~w k − ~v || = min {|| ~w s − ~v || : ~w s ∈ ∪ ∞ i =1 { ~w i }} ∞ s =1 and apply the magnetization Λ : C Int −→ V n and Λ : C Int −→ V n , then we obtainthe following paths Λ ∪ ∞ i =1 { ~t i } ( ~v ) := Λ ~t j ( ~v ) = ~t j − ~v with ~t j · ~v = 0 and Λ ∪ ∞ i =1 { ~w i } ( ~v ) := Λ ~w k ( ~v ) = ~w k − ~v with ~w k · ~v = 0. It follows that ~w k = λ~t j for some λ ∈ R . Since ~v was taken arbi-trarily and for any such choice there is a unique choice of magnet on the boundaryof each simple close curve minimizing the distance from ~v , the claim follows. (cid:3) T. AGAMA Application to Bellman’s lost in the forest problem
This section pretty much illustrates a sketch of an application of the underlyingnotion to Bellman’s lost in the forest problem. The solution is quite algorithmic innature but works primarily in parallel with the above developments. The Bellmanlost in the forest problem is one of the most important problems in the area ofoptimization, yet we find the following tools developed in the foregone sectionuseful. The problem is often stated in the following manner:
Question 2 (Bellman’s lost in the forest problem) . Given a hiker lost in the forestwith his orientation unknown, what is the best decision to be taken to exit in theshortest possible time taking into into consideration the shape of the boundary ofthe forest and the dimension of the space covering the forest?4.1.
A sketch partial solution.
First we classify the infinite collection of simpleclose curves in R n with magnetic boundaries according as they are isomorphic.That is, we consider the partition M := ∞ [ k =1 {C i ⊂ R n | C i ≍ C k } such that {C i ⊂ R n | C i ≍ C k } T {C i ⊂ R n | C i ≍ C j } = ∅ for k = j and C k and C j are not isomorphic. Let us pick arbitrarily a simple close curve C arbitrarily in M .Then it follows that C ∈ {C i ⊂ R n | C i ≍ C k } for some k ≥
1. The simple close curve C with magnetic boundary equipped withmagnets ∪ ∞ i =1 { ~u i } is uniquely determined upto constant dilates of magnets by virtueof Proposition 2.1. Now let ~v = ~O be an arbitrary point (hiker) lost in C Int (forest).Appealing to Theorem 2.5 we choose O ~v ( δ ) = min {O ~v ( ǫ ) | ~u j ∈ O ~v ( ǫ ) , ǫ > } (4.1)for each ~u j ∈ ∪ ∞ i =1 { ~u i } . By virtue of our choice there exists some magnet ~u t ∈O ~v ( δ ) for ~u t ∈ ∪ ∞ i =1 { ~u i } since magnets are dense on the boundary C B by virtue ofDefinition 2.2. Next we apply the magnetization Λ : C Int −→ V n and obtain theexit path Λ ∪ ∞ i =1 { ~u i } ( ~v ) = Λ ~u t ( ~v ) = ~u t − ~v with ~v · ~u t = 0 by appealing to Theorem 2.4 with the least measure of magnetization || ~v − ~u t || = min {|| ~v − ~u s || : ~u s ∈ ∪ ∞ i =1 { ~u i }} . Thus the hiker exits the forest without being privy to information of his orientationwithin the forest with the shortest path ~u t − ~v .It has to be said that the solution as espoused in the sketch may not be viewed as acomplete solution to the Bellman lost in the forest problem, because in practice theequivalence of the minimal distance of the hiker ~v to the boundary of the simpleclose curve (forest) C B with the notion magnetization Λ : C Int −→ V n under theadditional orthogonality condition of the hiker and some point on the boundary IMPLE CLOSE CURVE MAGNETIZATION AND APPLICATION TO BELLMAN’S LOST IN THE FOREST PROBLEM7 of the forest ~v · ~u t = 0 may not necessarily hold. Nonetheless we believe it is stillpossible the problem could be studied under the equivalenceΛ ~u ,~u ,...,~u k ( ~v ) = Λ ~u j ( ~v ) = ~u j − ~v if and only if || ~v − ~u j || = min {|| ~u s − ~v ||} ks =1 without the extra regime ~v · ~u j = 0 forany hiker in the forest. References
1. Bellman, Richard,
Minimization problem
Bull. Amer. Math. Soc, vol. 62:3, 1956, pp. 270.2. Finch, Steven R and Wetzel, John E ,
Lost in a forest
The American Mathematical Monthly,vol. 111:8, Taylor & Francis, 2004, pp. 645–654.3. Ward, John W
Exploring the Bellman Forest Problem , 2008.
Department of Mathematics, African Institute for Mathematical science, Ghana
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