A GENERALIZATION OF JUNG S THEOREM. M. Henk

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1 A GENERALIZATION OF JUNG S THEOREM M. Henk Abstract. The theorem of Jung establshes a relaton between crcumraus an ameter of a convex boy. The half of the ameter can be nterprete as the maxmum of crcumra of all 1-mensonal sectons or 1-mensonal orthogonal projectons of a convex boy. Ths pont of vew leas to two seres of j-mensonal crcumra, efne va sectons or projectons. In ths paper we stuy some relatons between these crcumra an by ths we fn a natural generalzaton of Jung s theorem. Introucton Throughout ths paper E enotes the -mensonal euclean space an the set of all convex boes K E compact convex sets s enote by K. The affne (convex) hull of a subset P E s enote by aff(p) (conv(p)) an m(p) enotes the menson of the affne hull of P. The nteror of P s enote by nt(p) an relnt(p) enotes the nteror wth respect to the affne hull of P. enotes the euclean norm an the set of all -mensonal lnear subspaces of E s enote by L. L enotes for L L the orthogonal complement an for K K, L L the orthogonal projecton of K onto L s enote by K L. The ameter, crcumraus an nraus of a convex boy K K s enote by D(K), R(K) an r(k), respectvely. For a etale escrpton of these functonals we refer to the book [BF]. Wth ths notaton we can efne the followng -mensonal crcumra Defnton. For K K an 1 let ) Rσ(K) := max max R(K (x + L)), L L x L ) R π(k) := max L L R(K L). We obvously have Rσ +1 (K) Rσ(K), Rπ +1 (K) Rπ(K), Rπ(K) Rσ(K) an Rσ(K) = Rπ(K) = R(K), Rσ(K) 1 = Rπ(K) 1 = D(K)/2. The theorem of Jung [J] states a relaton between the crcumraus an the ameter of a convex boy. On account of the efnton of Rσ(K), Rσ(K) 1 we can escrbe hs result as follows 1991 Mathematcs Subject Classfcaton. AMS 52A43. Key wors an phrases. Crcumraus, Dameter, Inraus. I woul lke to thank Prof. Dr. J. M. Wlls, who calle my attenton to these generalze crcumra 1 Typeset by AMS-TEX

2 2 M. HENK Theorem of Jung. Let K K. Then 2 Rσ(K) + 1 R1 σ(k), (1.1) an equalty hols f an only f K contans a regular -smplex wth ege length D(K). In the same way the theorem may be escrbe wth the crcumra R ρ(k) an R 1 ρ(k). Here we stuy n general the relatons between the -mensonal an j-mensonal crcumraus of these both seres an get the followng results Results Theorem 1. Let K K an 1 j. Then Rσ(K) (j + 1) j( + 1) Rj σ(k), (1.2) an equalty hols for > j f an only f K contans a regular -smplex wth ege length R j σ(k) 2(j+1) j. Theorem 2. Let K K an 1 j. Then Rπ(K) (j + 1) j( + 1) Rj π(k), (1.3) an equalty hols for > j f an only f an orthogonal projecton of K onto an - mensonal lnear subspace contans a regular -smplex wth ege length R j π(k) Let us remark that both theorems are a generalzaton of the classcal theorem of Jung snce for =, j = 1 the nequaltes (1.2) an (1.3) become (1.1). Proofs 2(j+1) j. To prove these theorems t s necessary to examne n more etal the crcumra of smplces snce the crcumraus of a convex boy K s etermne by the crcumraus of a certan smplex T K. Ths well known fact s escrbe n the followng lemma Lemma 1. Let K K an 0 be the center of the crcumball of K. Then there exsts a k-smplex T K, T = conv({x 0,...,x k }) wth 0 relnt(t), R(T) = R(K) an x = R(K), 0 k. Proof. cf. [BF], p. 9 an p. 54. Wth ths lemma t s easy to fn such ( 1)-mensonal planes for a smplex whch prouce the maxmal ( 1)-crcumraus wth respect to projectons or sectons.

3 A GENERALIZATION OF JUNG S THEOREM 3 Lemma 2. Let T K be a -smplex, ˆF a face of T wth maxmal crcumraus an ˆL L 1, ˆx ˆL wth ˆx + ˆL = aff( ˆF). Then ) Rσ 1 (T) = R(T (ˆx + ˆL)) = R( ˆF), ) Rπ 1 (T) = R(T ˆL) = R( ˆF). Proof. Let L 1 L 1 wth R 1 π (T) = R(T L 1 ) an let T L 1 the convex hull of the ponts x 0,...,x, where x 0,...,x enote the mages of the vertces of T uner the projecton onto L 1. Further let 0 be the center of the crcumball of T L 1 an T T L 1, T = conv({x 0,...,x k }), 1 k 1, a k-smplex wth the propertes of lemma 1. Now let F be a face of T contanng such k + 1 vertces whch are mappe onto x 0,...,x k wth respect to the orthogonal projecton onto L 1. We have R( ˆF) R(F) R(F L 1 ) R(T) = Rπ 1 (T); otherwse R( ˆF) Rσ 1 (T) Rπ 1 (T) an the asserton follows. On account of the lemma above we have R(S)/Rπ 1 (S) = R(S)/Rσ 1 (S) = /( 2 1) 1/2 for a regular -smplex S. That ths s even an upper boun for every smplex s shown n the next lemma. Lemma 3. Let T K a smplex. Then ) R(T) 2 1 R 1 (T), ) R(T) 2 1 R 1 (T), an equalty hols f an only f T s a regular -smplex. Proof. If T s a regular -smplex we have equalty by lemma 2. Hence on account of Rσ 1 (T) Rπ 1 (T) t suffces to prove the lemma for the ( 1)-crcumraus Rσ 1 (T). Let 0 be the center of the crcumball of T an {x 0,...,x k } a sutable subset of the vertces of T, such that T = conv({x 0,...,x k }) has the propertes of lemma 1. If k < then R(T) = R(T) = R 1 σ (T) < σ π 2 1 R 1 σ (T). (2.1) Hence we may assume that T = conv({x 0,...,x }) s a -smplex wth 0 nt(t) an x = R(T), 0. Let λ be the maxmal raus of a -mensonal ball wth center 0, whch s contane n T. Ths ball touches a face F of T n a pont λa, a = 1. Let F be gven by conv({x 1,...,x }). Snce a s a normal vector of aff(f) we have x λa 2 = R(T) 2 λ 2, 1. Hence λa s the center of the crcumball of F [BF, p. 54] an t follows R(T) 2 R 1 σ (T) 2 λ 2. (2.2)

4 4 M. HENK For the nraus r(t) of a smplex T we have r(t) R(T)/ [F] an so by the choce of λ λ 2 R(T)2 2. (2.3) Along wth (2.2) ths shows the nequalty ). If we have equalty n the relaton ) then from (2.1), (2.2) an (2.3) follows that T s a -smplex wth r(t) = R(T)/. Ths s only possble f T s regular [F]. Now we are able to prove the theorems. Proof of Theorem 1. It obvously suffces to show the nequaltes R σ(k) 2 1 R 1 σ (K), 1 <. (2.4) Snce the crcumra are nvarant wth respect to translatons we may assume that there s an -mensonal lnear subspace L L wth R σ(k) = R(K L ) an 0 s center of the crcumball of K L. Moreover let T (K L ) a k-smplex wth the propertes of lemma 1. Denotng by Rσ 1 (T;L ) the ( 1)-crcumraus of T wth respect to the euclean space L we get from lemma 3 R(T) 2 1 R 1 σ (T;L ). (2.5) By the choce of T we have R(T) = Rσ(K) an snce Rσ 1 (K) Rσ 1 (T;L ) the nequaltes (1.1) are shown. If an nequalty of (1.1) s satsfe wth equalty for > j we must have equalty n (2.4) an (2.5). By lemma 3 ths means that T s a regular -smplex whch satsfes the relaton R(T) = R σ(k) = (j + 1) j( + 1) Rj σ(k). (2.6) Snce T s regular we have R(T) = (/(2 + 2)) 1/2 D(T) an by (2.6) we see that T has the ameter (ege length) R j σ(k)((2j + 2)/j) 1/2. Now let T be a regular -smplex contane n K wth the gven ege length. On account of (1.1) we get R(T) = D(T) = (j + 1) j( + 1) Rj σ(k) R σ(k). (2.7) Clearly R(T) R σ(k) an so we can replace by = n (2.7). Proof of Theorem 2. On account of lemma 3 the proof can be one n the same way as the proof of theorem 1.

5 A GENERALIZATION OF JUNG S THEOREM 5 Remarks (1) If we replace the frst maxmum conton by a mnmum conton n the efnton of the crcumra we get two other seres of -crcumra whch now start wth the half of the wth of a convex boy. If we further replace the crcumraus by the nraus we totally get four seres of crcumra an four seres of nra. Some of these functonals are stue n Computatonal Geometry [GK]. For a survey of these generalze crcumra an nra we refer to [H]. (2) Theorems nvolvng nraus, crcumraus, ameter an wth have a long traton n the geometry of convex boes. In ths context we refer to [BL], [BF], [E], [DGK]. References [BF] T. Bonnesen, W. Fenchel, Theore er konvexen Körper, Sprnger, Berln, [BL] W. Blaschke, Kres un Kugel, Vet; Secon e., W. e Gruyter, Berln, Lepzg, [DGK] L. Danzer, B. Grünbaum an V. Klee, Helly s theorem an ts relatves. In Convexty (V. Klee, e.), Amer. Math. Soc. Proc. Symp. Pure Math. 13 (1963), [E] H.G. Eggleston, Convexty, Cambrge Unv. Press, Cambrge, 1958, [F] L. Fejes Tóth, Extremum propertes of the regular polytopes, Acta. Math. Aca. Sc. Hungar. (1955), [GK] P. Grtzmann an V. Klee, Inner an outer j-ra of convex boes n fnte-mensonal norme spaces, to appear n Ds. an Comp. Geometry (1991). [H] M. Henk, Unglechungen für sukzessve Mnma un verallgemenerte In- un Umkugelraen konvexer Körper, Dssertaton, Unverstt Segen (1991). [J] H.W.E. Jung, Über e klenste Kugel, e ene räumlche Fgur enschleßt, J. Rene Angew. Math. 123 (1901), Mathematsches Insttut, Unverstät Segen, Hölerlnstrasse 3, D-W-5900 Segen, Feeral Republc of Germany.