ON CONVEX BODIES OF CONSTANT WIDTH 4 0 L.E. BAZYLEVYCH AND M.M. ZARICHNYI 0 2 Dedicated to Professor E.D. Tymchatyn on the occasion of his 60th anniversary n a J Abstract. We present an alternative proof of the following fact: the hy- 7 perspace of compact closed subsets of constant width in Rn is a contractible Hilbert cube manifold. The proof also works for certain subspaces of com- ] pactconvexsetsofconstantwidth aswellas forthe pairsofcompactconvex G sets of constantrelativewidth. Besides,it is provedthat the projectionmap M of compact closed subsets of constant width is not 0-soft in the sense of . Shchepin, in particular, is not open. h t a m The topology of the hyperspace of compact convex sets in euclidean spaces [ was investigated by different authors; see, e.g. [1, 2, 3, 4]. 1 In this note we consider some topologicalproperties of (the maps of) compact v convex bodies of constant width. A convex set in euclidean space is said to be 0 6 of constant width d if the distance between two supporting hyperplanes equals 0 d in every direction. To be more formal, denote by h : Sn−1 R the support 1 function of a convex body K in Rn defined as follows: Kh (u) =→max x,u x 0 K 4 K . Here, as usual, , stands for the standard inner product in Rn{ahnd Sin|−1 ∈is 0 th}e unit sphere in Rnh.iThe widths function of K is the function w : Sn−1 R / K h defined by the formula w (u) = h (u) h ( u). A convex body K →is of t K K − K − a constant width d > 0 provided w is a constant function taking the value d. K m It was proved by the first-named authors [3] that the hyperspace of compact : v convex bodies of constant (non-specified) width in euclidean space of dimension i X 2 is homeomorphic to the punctured Hilbert cube. In Section 1 we present a ≥ r more direct proof of this result. The technique allows also to prove that some a subspaces of the above mentioned hyperspace as well as the hyperspace of pairs of compact convex sets of constant relative width are manifolds modeled on the Hilbert cube (Q-manifolds). In connection to the topology of the hyperspace of compact convex sets of nonmetrizable compact subsets in locally convex spaces it was proved in [5] that, for any affine continuous onto map of convex subsets in metrizable locally convex spaces, the natural map of the hyperspaces of compact convex subsets is soft in the sense of Shchepin [6]. In Section 2 we demonstrate that this is not 1991 Mathematics Subject Classification. 54B20, 52A20, 46A55. Key words and phrases. Convex body, constant width, Q-manifold, soft map. 1 2 L.E. BAZYLEVYCH ANDM.M. ZARICHNYI the case if we restrict ourselves with the compact convex sets of constant width; the map under consideration is not even open. 1. Hyperspaces of compact convex bodies of constant width By cc(Rn) we denote the hyperspace of compact convex subsets in Rn. We equip cc(Rn) with the topology generated by the Hausdorff metric. By cw (Rn) we denote the subset of cc(Rn) consisting of convex bodies of d constant width d > 0 in Rn. We also put cw (Rn) = x x Rn . In [3] it 0 is proved that the hyperspace cw(Rn) = cw (Rn){,{n } |2, i∈s a co}ntractible d>0 d ∪ ≥ Hilbert cube manifold. Here we essentially simplify the proof of this result. Theorem 1.1. Let D [0, ) be a convex subset such that D (0, ) = . The hyperspace cw (R⊂n) ∞d D is a contractible Hilbert cube∩man∞ifold6. ∅ d ∪{ | ∈ } Proof. First, we embed cw(Rn) as a convex subset of a Banach space. Define a map ϕ: cw(Rn) C(Sn−1) by the formula ϕ(K)(x) = h (x), K K cw(Rn). It is a well-kno→wn fact that ϕ is a continuous map. Moreover, ∈ it is obvious that ϕ is an embedding which is an affine map in the sense that ϕ(tA+(1 t)B) = tϕ(A)+(1 t)ϕ(B) for every A,B cw(Rn) and t [0,1]. The image−of ϕ is a locally com−pact convex subset of C∈(Sn−1). ∈ We are going to prove that, for any d > 0, the space cw(Rn) is infinite- dimensional. First consider the case n = 2. Let K denote the Reuleaux triangle in R2 that is the intersection of the closed balls of radius d centered at (0,0), (d,0), and (d/2,d√3/2). For any α [0,2π], denote by K the convex body obtained by α ∈ rotation of K by angle α conterclockwise around the origin. We show that the theset K α [0,2π) containslinearlyindependent subset ofarbitraryfinite α { | ∈ } cardinality. To this end, one has to demonstrate that the family of the support functions h : S1 R contains linearly independent subset of arbitrary finite Kα → cardinality. We identify S1 with the set eit t [0,2π] . It is easy to see that h (eiγ) = h (ei(γ−α)). Elementary geom{etric| ar∈guments}demonstrate that Kα K h [0,π/3] = 1, h [π,4π/3] = 0, h ((π/3,π) (4π/3,2π)) (0,1). K K K | | ∪ ⊂ Fix a natural number k. For each j = 0,1,...,k 1, let h = h . j K − (jπ)/(3k) In order to demonstrate that the functions h , j = 0,1,...,k, are linearly j k independent, consider a linear combination g = λ h . Suppose that g = 0, Pj=0 j j ON CONVEX BODIES OF CONSTANT WIDTH 3 then k k g(π/3) = Xλjhj(π/3) = Xλj = 0, j=0 j=0 g((π/3)+(π/(3k))) = λ h ((π/3)+(π/(3k))) 0 0 k +Xλjhj((π/3)+(π/(3k))) = 0, j=1 whence λ = 0. Consequently evaluating the function g at the points (π/3)+ 0 (jπ/(3k)), j = 2,...,k, we conclude that λ = 0 for every i = 0,1,...,k. i In the case n > 2, consider the family of n-dimensional simplices in Rn all whose edges are of length d. For every such a simplex, ∆, consider the intersection of all closed balls of radius d centered at the vertices of ∆ and containing ∆. The obtained set, L, is obviously a compact convex subset in Rn of constant width d. Denote by the family of all compact convex subsets in Rn that can be obtained in this Lway. Let pr: Rn R2 denote the projection. This projection generates the map cc(pr): cc(Rn) →ccRm, f(A) = pr(A). The → map cc(pr) is an affine map. ′ ′ Then cc(pr)(L) L K α [0,2π] and we conclude that the α space {and, therefo|re cw∈(LR}n)⊃is{infin|ite-d∈imensio}nal. d L To finish the proof, apply the results on topology of metrizable locally com- pact convex subsets in locallyconvex spaces [8]. Since the space cw(Rn) is easily shown to be locally compact, the Keller theorem (see [8]) implies that cw(Rn) is a Q-manifold. (cid:3) Corollary 1.2. Let d 0. The hyperspace cw (Rn) d d is homeo- 0 d 0 ≥ ∪{ | ≥ } morphic to a punctured Hilbert cube Q . \{∗} Proof. By a result of Chapman [9], it is sufficient to prove that there exists a proper homotopy H: cw (Rn) d d [1, ) cw (Rn) d d . d 0 d 0 ∪{ | ≥ }× ∞ → ∪{ | ≥ } This homotopy can be defined in an obvious way, H(K,t) = tK, (K,t) cw (Rn) d d [1, ). (cid:3)∈ d 0 ∪{ | ≥ }× ∞ Theorem 1.3. Let X be a convex subset in Rn that contains a closed square of side d. Then for any convex subset D of [0,d] with D (0,d] = the set cw (X) = cc(X) cw (Rn) is a Hilbert cube manifold. ∩ 6 ∅ D D ∩ Proof. Note that the family L x L cw (X), x X contains the D { − { } | ∈ ∈ } family K α [0,2π) defined in the proof of Theorem 1.1 (for fixed d). α { | ∈ } Since X is a subset of a finite-dimensional linear space and K α [0,2π) α { | ∈ } 4 L.E. BAZYLEVYCH ANDM.M. ZARICHNYI contains an infinite linearly independent family, we conclude that cw (X) also D contains an infinite linearly independent family. Therefore, the space cw (X) D is infinite-dimensional. Then we apply the arguments of the proof of Theorem (cid:3) 1.1 Remark that it follows, in particular, from Theorem 1.3 that the hyperspace of convex bodies that can be rotated inside a square is homeomorphic to the Hilbert cube, because this hyperspace is a compact contractible Q-manifold(see [7]). 2. On softness of the projection map A map f: X Y is soft (respectively n-soft) if for every commutative dia- → gram ψ // A X i f (cid:15)(cid:15) (cid:15)(cid:15) Z // Y, ϕ where i: A Z is aclosed embedding into a paracompact space Z (respectively → a paracompact space Z of covering dimension n), there exists a map Φ: Z ≤ → X such that Φ A = ψ and fΦ = ϕ. The notion of (n-)soft map was introduced | by E.V. Shchepin [6]. Let pr: Rn Rm denote the projection, n m. As we already remarked, this projection→generates the map cc(pr): cc(R≥n) ccRm and by → p = cw(pr): cw(Rn) cwRm → we denote its restriction onto cc(pr) cw(Rn). It is proved in [5] that the map | cc(pr) is soft. The images of the fibers of the map p under the embedding ϕ are obviously convex. It is natural to ask whether the map p is also soft. As the following result shows, the answer turns out to be negative. The idea of the proof is suggested by S. Ivanov. Theorem 2.1. The map p: cw(R3) cwR2 is not 0-soft. → Proof. Consider the compactum L in R3 which is the intersection of the closed balls of radius 2 centered at the points (0,1,0), (0, 1,0), ( 1,0,√2, and − − (1,0,√2 (a Reuleaux tetrahedron). It is well-known that L is of constant width and f(L) = K, where K denotes the disc of radius 1 centered at the origin in R2. For every i, denote by K the compactum in R2 described as follows. Let i x = (cos(2πj/(2i+1)),sin(2πj/(2i+1))), j = 0,1,...,2i. j ON CONVEX BODIES OF CONSTANT WIDTH 5 The compactum K is the intersection of the discs of radius x x centered i 0 i at the points xj, j = 0,1,...,2i. Obviously, Ki cwR2 andklimi−→∞Kki = K in ∈ the Hausdorff metric. Let S = 0 1/n n N , f: 0 cw(R3) be the map that sends 0 into L, and F:{S }∪{cw(R2|) b∈e th}e ma{p }su→ch that F(0) = K and F(1/n) = K , n n N. → ∈Supposenowthatthemappis0-soft. Thenthereexist amapG: S cw(R3) → such that G(0) = f(o) = L and pG = F. Denoting G(1/i) by L , we see that i p(Li) = Ki and limi→∞Li = L. Since every diameter of a convex body (i.e., a segment that connects the points at which two parallel supporting hyperplanes touch the body) of constant width is orthogonal to the supporting planes, for every i, there exists a plane in R3 containing the set pr−1(∂K ) L . From this i i we conclude that there exists a plane in R3 containing the set ∩ pr−1(∂K) L (0,1,0),(0, 1,0),( 1,0,√2,(1,0,√2 , ∩ ⊃ { − − } (cid:3) a contradiction. Corollary 2.2. The map f: cw(R3) cwR2 is not open. → 3. Pairs of compact convex bodies of constant relative width Two convex bodies K ,K Rn are said to be a pair of constant width 1 2 if K K is a ball (Maehara⊂[10]). We denote by crw(Rn) the set of all 1 2 pairs o−f compact convex bodies of constant relative width. The set crw(Rn) is topologized with the subspace topology of cc(Rn) cc(Rn). Note that the spacecw(Rn) canbenaturallyembedded into crw(Rn)×bymeans ofthemapping A (A,A). 7→ Theorem 3.1. The space crw(Rn) is a contractible Q-manifold. Proof. First, note that the map β: (A,B) (h ,h ) affinely embeds the space A B crw(Rn) into C(Sn−1) C(Sn−1) and the7→image of this embedding is a convex subset of C(Sn−1) C(×Sn−1). Since the space cw(Rn) is infinite-dimensional, so is crw(Rn). It is ea×sy to see that cw(Rn) is locally compact. Arguing like in the proofofTheorem1.1, weconcludethatcw(Rn)isacontractibleQ-manifold. (cid:3) 4. Remarks and open questions The set cw (Rn) is the preimage ofthe set of closed balls of radius d under the d central symmetry map c: cc(Rn) cc(Rn), c(A) = (A A)/2. Since cw (Rn) d → − is an absolute retract, this directly leads to the following question. Question 4.1. Is the central symmetry map soft as a map onto its image. Let L be a Minkowski space (i.e. a finite-dimensional Banach space) of di- mension 2. Given a compact convex body K L, one says that K is of ≥ ⊂ 6 L.E. BAZYLEVYCH ANDM.M. ZARICHNYI constant width if the set K K is a closed ball centered at the origin. A natu- − ral question arizes: for which spaces L the results of this paper can be extended over the hyperspaces of convex bodies of constant width in L? Note that for the space (Rn, ∞) the hyperspace of convex bodies of constant width coincides k·k with that of closed balls in it and therefore is finite-dimensional. In addition, a counterpart of Theorem 2.1 does not hold for this space. Question 4.2. Describe the topology of the hyperspace of smooth convex bod- ies of constant width. Every compact convex body of constant width is a rotor in cubes as well as in some other polyhedra. Question 4.3. Find counterparts of the results of this paper for rotors in an- other polyhedra (e.g., equilateral triangles). References [1] S.B. Nadler, Jr., J. Quinn, N.M. Stavrokas,Hyperspace of compact convex sets, Pacif. J. Math. 83(1979), 441–462. [2] L. Montejano, The hyperspace of compact convex subsets of an open subset of Rn, Bull. Pol. Acad. Sci. Math. V.35, No 11-12(1987),793–799. [3] Bazylevych L.E. Topology of the hyperspace of convex bodies of constant width, Mat. zametki. 62 (1997), 813–819(in Russian). [4] BazylevychL.E.On the hyperspace of strictly convex bodies, Mat.studii.2(1993),83–86. [5] M. Zarichnyi, S. Ivanov, Hyperspaces of compact convex subsets in the Tychonov cube, Ukr. mat. zh. 53 (2001), N 5, 698–701;translation in Ukrainian Math. J. 53 (2001), no. 5, 809–813. [6] E. V. Shchepin, Functors and uncountable powers of compacta. Uspekhi Mat. Nauk 31 (1981), 3-62. [7] T. Chapman, Lectures on Hilbert cube manifolds, CBMS Regional Conference Series in Math. 28(1976). [8] Bessaga C., Pe lczyn´ski A. Selected topics in infinite-dimensional topology.- Monografie Matematyczne, 58, Warsaw: PWN, 1975. [9] Chapman T.A. On some applications of infinite-dimensional manifolds to the theory of shape, Fund. Math. 76 (1972), 181–193. [10] H.Maehara,Convexbodies formingpairs ofconstantwidth,J.Geom.22(1984),101–107. NationalUniversity“LvivPolytechnica”,12BanderyStr.,79013Lviv,Ukraine Lviv National University, 1 Universytetska Str., 79000 Lviv, Ukraine E-mail address: [email protected]