THE WILLS CONJECTURE

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 349, Number, October 997, Pages S THE WILLS CONJECTURE NOAH SAMUEL BRANNEN Abstract. Two stregthegs of the Wlls cojecture, a exteso of Boese s radus equalty to -dmesoal space, are obtaed. Oe s the sharpest of the kow stregthegs of the cojecture three dmesos; the other employs techques whch are fudametally dfferet from those utlzed the other proofs. Oe of the best kow geometrc equaltes s the sopermetrc equalty, whch states that of all closed plaar curves wth fxed permeter, a crcle ecloses the greatest area. If K saplaarcovexbodyacovex body sacompact covex set wth oempty teror of permeter L wth area A, the ths equalty ca be expressed as L 2 4πA wth equalty oly whe the boudary of K s a crcle. A stregtheg of was gve by the Dash mathematca T. Boese, who demostrated 929 see 5] that f K has crcumradus R ad radus r, the 2 L 2 4πA π 2 R r 2. Ths equalty ca be derved from the equalty 3 A λl + λ 2 π, r λ R. The equalty 3 wth λ = r s kow as Boese s radus equalty. J. M. Wlls 5] cojectured 97 that 4 V rs +r ω, whch would be a exteso of Boese s radus equalty to hgher dmesos. Here V s the -dmesoal volume of a covex body R, r s ts radus, S s ts -dmesoal surface area ad ω s the volume of the -dmesoal ut ball. Ths result was prove smultaeously 973 by J. Bokowsk 2] ad V. I. Dskat 7]. Both proofs demostrated that the Wlls cojecture s a cosequece of a result due to H. Hadwger 8], amely S ω 5 V S rω ]. Receved by the edtors February 9, 995 ad, revsed form, August 5, Mathematcs Subject Classfcato. Prmary 52A4. Key words ad phrases. Crcumradus, covex body, er parallel body, radus, mxed volume, quermasstegral c 997 Amerca Mathematcal Socety Lcese or copyrght restrctos may apply to redstrbuto; see

2 3978 NOAH SAMUEL BRANNEN I 979 Robert Osserma ] showed that V rs +r S 6 ω, r ω whch s a stregtheg of the Wlls cojecture. He derved ths result from 5. I 988 Jae Sagwe-Yager ] derved a eve stroger result, V rs +r 2 W 2 + r sv K s,...,k s,k r,b,bds, where W 2 s the secod quermasstegral of K. Here K s ad K r are er parallel bodes of K, advk s,...,k s,k r,b,bsamxedvolume. The pertet deftos wll be gve subsequetly. Sagwe-Yager derved her result as a corollary to a lower boud o the volume of er parallel bodes. We shall gve aother lower boud o the volume of er parallel bodes. Ths result has as a corollary V rs + tv K t,...,k t,k,...,k,b,bdt, = }{{} a stregtheg of the Wlls cojecture whch s a sharper equalty R 3 tha the equalty gve by Sagwe-Yager. It also provdes the shortest proof of the Wlls cojecture that we have bee able to fd. A addtoal stregtheg of the Wlls cojecture s V rs + = + ] r + W + K r +r ω, where W + s the +st quermasstegral. Ths stregtheg results from takg a approach whch s fudametally dfferet tha the methods employed the other proofs. We beg wth some deftos. If K ad L are covex bodes, ther Mkowsk sum K + L s defed to be {k + l : k K, l L}. If λ s a scalar the λk s {λk : k K}. If the covex body K R s a Mkowsk lear combato of m covex bodes,.e. K = λ K + λ 2 K λ m K m, λ,...,λ m, the the volume of K ca be expressed as a th degree homogeeous polyomal the λ as follows: V K = V K p,k p2,..., K p λ p λ p2...λ p. p,...,p m Here the summato s exteded over all p depedetly as vares from to see 3], page 275. The coeffcets V K p,k p2,..., K p are called mxed volumes. The mxed volume V K,...,K,B,...,B wth copes of the ut ball B R wll be deoted by W K ad s called the th quermasstegral of K. It ca be show that W K =VK, W K =SK, 2 W K=MK, ω Lcese or copyrght restrctos may apply to redstrbuto; see

3 THE WILLS CONJECTURE 3979 where MK s the mea wdth of K, ad W K=ω. More geerally, for a fxed covex body E we defe W K; E =VK,...,K,E,...,E to be the th relatve quermasstegral of K. Also, we defe the relatve radus of K wth respect to E to be rk; E =sup{r : some traslate of re K}, ad the relatve surface area of K wth respect to E to be SK; E =W K; E. Mxed volumes have several terestg propertes. We preset four propertes whch wll be used the stregthegs of the Wlls cojecture. Property. V,,..., s symmetrc ts argumets. Property 2. V,,..., s a Mkowsk multlear fuctoal. Property 3. V,,..., s mootoe, the sese that f K K,the VK,K 2,..., K V K,K 2,..., K. Property 4. If K,...,K are covex bodes R ad m<s a atural umber, the m V K,...,K m V K,...,K,K }{{} m+,...,k. = m The above equalty s due to Alexadrov ]. Proofs of the frst three propertes ca be foud 3], pages The followg formula cocerg the quermasstegrals of Mkowsk sums of covex bodes wll be used a stregtheg of the Wlls cojecture. If K ad L are covex bodes R, E safxedcovexbodyr ad s ad t are scalars, the the th relatve quermasstegral of sk + tl satsfes 7 W sk + tl = s j t j V K,...,K,L,...,L,E,...,E. j }{{} j= j j Ths formula s a geeralzato of a result kow as Steer s formula Steer demostrated ths result oly for the case = 3, whch s 8 V K + λb = λ W K. = We tur our atteto ow to er parallel bodes. I the followg, all our cosderatos wll be restrcted to -dmesoal Eucldea space R, 2. The outer parallel body at a dstace λ of a covex body K wll be deoted by K λ = K + λb, wherebs the ut ball. The er parallel body of K at a dstace λ, whch s the set of all pots of K whose dstace from the boudary of K s at least λ, wll be deoted by K λ. Lcese or copyrght restrctos may apply to redstrbuto; see

4 398 NOAH SAMUEL BRANNEN We defe the relatve er parallel body of K wth respect to a fxed covex body E at a dstace λ, λ rk; E, to be K λe = {x : x + λe K}. Oce the covex body E s fxed we wll deote the relatve er parallel body smply by K λ. The ext two lemmas wll be used ofte what follows. The frst lemma s a drect cosequece of the above defto. Lemma. Let E be a fxed covex body ad let K be ay covex body R.Let the relatve radus of K be r. Foreachλ, λ r, K λ +λe K. Lemma 2. Uder the same assumptos as the prevous lemma, for each λ, λ r, let SK λ be the relatve surface area of K λ.the VK= SK λ dλ. The precedg lemma, as well as the ext, s due to Bol 4]. Lemma 3. W K te,e] s a cocave fucto of t for =,...,. We wll ow gve two lower bouds ad oe upper boud for the volume of relatve er parallel bodes. The two lower bouds wll be compared. Theorem. Sagwe-Yager If E s a fxed covex body ad K s ay covex body -dmesoal Eucldea space R ad K has volume V, relatve surface area S, relatve secod quermasstegral W 2, ad relatve radus r, the for all λ, λ r, V K λ V λs +λ 2 W λ λ sv K s,...,k s,k λ,e,eds. Ths lower boud for the volume of the relatve er parallel body at a dstace λ s due to Sagwe-Yager ]. What follows s a geeral result o the relatve quermasstegrals of relatve er parallel bodes. Theorem 2. If E s a fxed covex body ad K s ay covex body R wth th relatve quermasstegral W, +th relatve quermasstegral W + ad relatve radus r, the for all λ r, we have W K λ W λw + λw + K λ 2 λ V K λ,...,k λ,k,...,k,e,...,e. }{{} k= k k + Lcese or copyrght restrctos may apply to redstrbuto; see

5 THE WILLS CONJECTURE 398 Proof. By Lemma, K K λ + λe for all λ r. Usg ths fact ad the propertes of mxed volumes, we have W = V K,...,K,E,...,E V K λ + λe,k,...,k,e,...,e = V K λ,k,...,k,e,...,e+λw + V K λ,k λ +λe,k,...,k,e,...,e+λw + = V K λ,k λ,k,...,k,e,...,e 2 } {{ }} {{ } 2 +λv K λ,k,...,k } {{ } 2,E,...,E+λW }{{} W K λ +λw + + λw + K λ +λ V K λ,...,k λ,k,...,k,e,...,e. }{{} k= k k + Ths completes the proof. Wth SK deotg the relatve surface area of K wth respect to E, wehave the followg corollares. Corollary. Whe =Theorem 2 becomes V K λ V K λ SK λ SK λ λ V K λ,...,k λ,k,...,k,e, k= k k whch gves us a upper boud for the volume of K λ. The ext corollary gves us a lower boud for the volume of K λ. Corollary 2. For all λ r ad for all >2, VK λ VK λsk+ λ2 2 W 2K + k= λ tv K t,...,k t,k,...,k,e,edt. k k 2 Proof. Whe = Theorem 2 becomes SK λ SK λw 2K λw 2 K λ 3 λ V K λ,...,k λ,k,...,k,e,e. k= k k 2 Itegratg both sdes from to t wth respect to λ ad applyg Theorem 2, we obta V K V K t] t SK t2 2 W t 2K λv K λ,...,k λ,k,...,k,e,edλ, k= k k 2 whch whe rewrtte gves the corollary. Lcese or copyrght restrctos may apply to redstrbuto; see

6 3982 NOAH SAMUEL BRANNEN Theorem ad Corollary 2 to Theorem 2 gve lower bouds for the volume of the relatve er parallel body K λ. Whch equalty s sharper? For >3ths has ot bee determed, but we wll show that Corollary 2 s sharper whe =3. Whe = 3, Theorem s 9 V K λ V λs +2λ 2 W 2 +λ 2 W 2 K λ ad Corollary 2 to Theorem 2 s V K λ V λs λ2 W 2 +3 λ tw 2 K t dt. By Lemma 3, W 2 K t s a cocave fucto of t. Thus for all t satsfyg t λ, tw 2 K t t t W 2 K+ t ] λ λ W 2K λ =W 2 t+ λ W 2K λ W 2 ]t 2. Ths mples that λ tw 2 K t dt 2 λ2 W λ2 W 2 K λ W 2 ]= 6 λ2 W λ2 W 2 K λ. Substtutg the estmate to gves V K λ V λs λ2 W λ2 W 2 + ] 2 3 λ2 W 2 K λ, whch whe smplfed s 9. Ths shows that the rght had sde of s greater tha the rght had sde of 9, ad therefore s a stroger equalty. I 978 G. Mathero 9] showed the followg: Theorem 3. If K s a covex body R 2 wth area A, permeter L, ad radus r, the for all λ such that λ r oe has 3 AK λ A λl + πλ 2. He cojectured that R 3 would exted to 4 V K λ λ W K = for all covex bodes K R wth radus r ad for all λ satsfyg λ r. 4, f true, would be a Steer-lke formula for the volume of er parallel bodes. Sagwe-Yager showed 2] that 4 does ot hold R 3. However, whe =2adE=Bboth Theorem ad Corollary 2 to Theorem 2 become Mathero s equalty 3. Therefore both ca be vewed as extesos of 3 to R. The followg s a ope cojecture, due to Tesser 4]. Cojecture. If a... a are the real parts of the roots of λ W K, = where K s a covex body R ad λ R,the<a r R a,wherer ad R are the radus ad crcumradus of K, respectvely. Lcese or copyrght restrctos may apply to redstrbuto; see

7 THE WILLS CONJECTURE 3983 We meto ths cojecture because t s related to Boese s equalty, ad because Wlls s the oly perso to have made progress towards a proof see 6]. We ow cosder the stregthegs of the Wlls cojecture. Whe E = B ad λ = r Theorem becomes V rs +r 2 W 2 5 r + 2 r sv K s,...,k s,k r,b,bds. Sce mxed volumes are mootoe ad rb K, we kow that W 2 K =VK,...,K,B,B VrB,...,rB,B,B=r ω, whch shows that 5 s a stregtheg of the Wlls cojecture. Aga lettg E = B ad λ = r, Corollary 2 to Theorem 2 becomes 6 V K rsk+ tv K t,...,k t,k,...,k,b,bdt. = 2 Sce r tb K t ad rb K, wehave VK rsk+ tv r tb,...,r tb,rb,...,rb,b,bdt = 2 = V K rsk+ = tr t r V Bdt = V K rsk+ω r tr t dt = = V K rsk+ω r = r =VK rsk+ω r ++2, = or V K rsk+ω r, whch whe smplfed s the Wlls cojecture. We ca see that 6 s aother stregtheg of the Wlls cojecture, sce VK t,...,k t,k,...,k,b,b V r tb,...,r tb,rb,...,rb,b,b, 2 2 by the mootocty of mxed volumes. A addtoal stregtheg of the Wlls cojecture s a corollary to the followg theorem. ] Lcese or copyrght restrctos may apply to redstrbuto; see

8 3984 NOAH SAMUEL BRANNEN Theorem 4. If E s a fxed covex body ad K s ay covex body R wth volume V, relatve surface area S, relatve radus r, adw + K r deotes the relatve +st quermasstegral of the relatve er parallel body K r,the V rs + = + r + W + K r Proof. By Lemma, K K λ + λe. Therefore we have 7 S SK λ + λe. If we let S λ = SK r +r λe, ad S = SK r + re =SK r +r λe]+λe, the by 7 we have S 8 = λ W + K r +r λe, ad 9 = SK λ + λe = = Therefore by 7 ad 9, we have 2 SK λ S = = ad by 8 we have S λ = 2 S λ W + K λ. λ W + K λ, ] +r ω. λ W + K r +r λe. Aga by Lemma, K λ K λ r λ +r λe =K r +r λe,adso W K λ W K r +r λe for all =,...,. Ths fact, whe combed wth 2 ad 2, gves us S λ + S S = S = = S λ W + K r +r λe λ W + K λ SK λ. Itegratg both sdes of the above equalty from to r wth respect to λ, we have 22 by Lemma 2. S λ + S S dλ SK λ dλ = V, Lcese or copyrght restrctos may apply to redstrbuto; see

9 THE WILLS CONJECTURE 3985 Aother expresso for S λ,by7,s S λ =SK r+r λe= ad also = S = SK r + re = = r λ W + K r, r W + K r. Substtutg these expressos for S λ ad S to 22, we have V S + r λ W + K r r W + K r dλ = = S + = = = rs + = rs = Ths completes the proof. = + = r λ r ] W +K r dλ + r + W + K r r + W + K r Corollary. Whe E = B, the equalty ] V rs + r + + W + K r s a stregtheg of the Wlls cojecture sce ] r + W + + K r s oegatve. = ] r ω. +r ω How do these stregthegs compare to Osserma s stregtheg 6? Note that whe = 2adE = B, all four stregthegs become Boese s radus equalty A rl + πr 2. We wll frst show that 5 s a sharper equalty tha 6. If we apply Property 4 of mxed volumes to V B,K,...,K,B,theweseethat VB,K,...,K,B VB,...,BVK,...,K,B, whch s the same as W 2 K ω W K. Sce W K = SK,wehave SK 23 W 2 K ω. ω Lcese or copyrght restrctos may apply to redstrbuto; see

10 3986 NOAH SAMUEL BRANNEN If we rewrte 6 as V K rsk+r 2 ω SK ω, the by 23 we ca see that the rght had sde of 5 s greater tha the rght had sde of 6, whch meas that 5 s a sharper equalty. Whe = 3, Corollary 2 to Theorem 2 s a stroger equalty tha Theorem for all λ r. Therefore 6 s a sharper equalty tha 5 whe =3. I other words, V rs r2 W 2 +3 tw 2 K t dt s a stroger equalty tha 24 V rs +2r 2 W 2 +r 2 W 2 K r. Aga, for >3 t s ot kow whch equalty s stroger. Theorem 4 wth =3s 25 V rs +3r 2 W 2 K r πr3. Sce K r + rb K Lemma, by the mootocty of mxed volumes we have W 2 K r + rb W 2.Thus W 2 VK r +rb,b,b=w 2 K r +rv B, or r 2 W 2 r 2 W 2 K r πr3. Therefore 24 s a stroger equalty tha 25, whch tur mples that 6 s the strogest of the four stregthegs of the Wlls cojecture whe =3. We have see that Boese s radus equalty ca be exteded to hgher dmesos may ways. The extesos gve here volve volume, surface area, terms wth tegrals of certa mxed volumes, quermasstegrals of er parallel bodes, ad the radus. The Wlls cojecture has bee show to be false for >2 whe the radus s replaced wth the crcumradus; however, Bokowsk ad Hel 3] obtaed a smlar result wth the equalty reversed. Chakera 6] derved a Boese-style equalty whch yelds a stregtheed form of the sopermetrc equalty two dmesos, but whch does ot volve volume ad surface area hgher dmesos. For more formato o Boese-style equaltes ad related ope questos, see 3], pages Refereces. A. D. Aleksadrov, O the theory of mxed volumes of covex bodes, Mat. Sb , o. 45, Russa. 2. J. Bokowsk, Ee verschärfte uglechug zwsche Volume, Oberfläche ud Ikugelradus m R m,elem.math , MR 48:79 3. J. Bokowsk ad E. Hel, Itegral represetatos of quermasstegrals ad boese-style equaltes, Arch. Math , MR 88b: G. Bol, Bewes eer Vermutug vo H. Mkowsk, Abh. Math. Sem. Uv. Hamburg 5 943, MR 7:474f 5. T. Boese, Les problèmes des sopérmetres et des sépphaes, Gauther-Vllars, Pars, G. D. Chakera, Hgher dmesoal aalogues of a sopermetrc equalty of Beso, Mathematsche Nachrchte 48 97, MR 44:4643 Lcese or copyrght restrctos may apply to redstrbuto; see

11 THE WILLS CONJECTURE V. I. Dskat, A geeralzato of Boese s equaltes, Sovet Math. Doklady 4 973, o. 6, traslato of Doklady Akad. Nauk SSSR , MR 49: H. Hadwger, Vorlesuge über Ihalt, Oberfläche ud Isopermetre, Sprger, Berl, 957. MR 2:56 9. G. Mathero, La formule de Steer pour les érosos, Joural of Appled Probablty 5 978, MR 58:2874. R. Osserma, Boese-style sopermetrc equaltes, Amerca Math. Mothly , 29. MR 8h:523. J. R. Sagwe-Yager, Boese-style equaltes for Mkowsk relatve geometry, Trasactos of the Amerca Mathematcal Socety , o., MR 89g:527 2., A Boese-style radus equalty 3-space, Pacfc Joural of Mathematcs , MR 89f: R. Scheder, Covex bodes: The Bru-Mkowsk theory, Cambrdge Uversty Press, Cambrdge, 993. MR 94d: B. Tesser, Boese-type equaltes algebrac geometry, Semar o Dfferetal Geometry Prceto, Prceto Uversty Press, 982, pp MR 83d:52 5. J. M. Wlls, Zum Verhalts vo Volume zur Oberfläche be kovexe Körper, Arch.Math 2 97, MR 43: , Mkowsk s successve mma ad the zeros of a covexty fucto, Moatsh. Math. 9 99, MR 9f: Tama-cho, Fuchu-sh, Tokyo 83, Japa E-mal address: b-oah@hoffma.cc.sopha.ac.jp Lcese or copyrght restrctos may apply to redstrbuto; see