H metaptuqiak aut ergas a katat jhke sto Tm ma Majhmatik n tou Panepisthm ou Kr thc ton M rtio tou Epibl pwn tan o A. Giann pouloc. Thn epitrop

Transcription

1 ISOPERIMETRIKES ANISOTHTES KAI SUGKENTRWSH TOU METROU MARIA-ISABELLA UMWNOPOULOU DIPLWMATIKH ERGASIA TMHMA MAJHMATIKWN MARTIOS

2 H metaptuqiak aut ergas a katat jhke sto Tm ma Majhmatik n tou Panepisthm ou Kr thc ton M rtio tou Epibl pwn tan o A. Giann pouloc. Thn epitrop axiol ghshc apot lesan oi: A. Giann pouloc, M. Papadhmhtr khc kai S. Papadopo lou.

3 P mpto bibl o thc Sunagwg c tou P ppou tou Alexandr wc. O Je c, axi time Megej wn, dwse stouc anjr pouc thn idi thta na katanoo n th sof a kai ta majhmatik se risto kai telei tato bajm, dwse mwc kai se orism na ap ta z a en m rei aut n thn ikan thta. Stouc anjr pouc, pou qoun to q risma tou l gou, dwse thn ikan thta na k noun ta p nta s mfwna me touc kan nec thc logik c kai thc ap deixhc, en sta up loipa z a dwse to q risma na k noun m no,ti e nai qr simo kai wf limo gi th zw tou k je e douc, s mfwna me k poia pr noia thc f shc. Aut mpore kane c na to parathr sei se poll e dh z wn, kai id wc stic m lissec. E nai pr gmati axioja masth h org nws touc kai h peijarq a touc stic arqhgo c twn koinwni n touc, kai ak mh pi axioja masth h filotim a kai h kajari thta me thn opo a sull goun to m li kai h front da kai h t xh me thn opo a to apojhke oun. Giat afo, pwc x roume, oi jeo touc qoun anaj sei thn apostol na prosf roun stouc proikism nouc ap tic mo sec anjr pouc aut to komm ti ambros ac, eke nec den katad qthkan na to enapoj toun pou t qei sto q ma sto x lo, se opoiod pote llo morfo kai akat rgasto ulik, all, sull gontac ta eugen stera ulik ap ta wrai tera njh pou futr noun sth g, kataskeu zoun doqe a gi thn apoj keush tou melio, tic leg menec khr jrec, pou e nai lec sec kai moiec metax touc kai qoun sq ma ex gwno. Kai ja do me am swc oti aut to mhqane ontai akolouj ntac nan gewmetrik kan na. Aut pou endi fere kur wc tic m lissec tan na sqhmat zoun doqe a pou na e nai to na d pla sto llo kai na ef ptontai kat tic pleur c, ste na mhn eisqwro n sta mesodiast mata x na s mata pou na katastr foun to m li. Aut mporo san na to pet qoun me tr a euj gramma sq mata, ki enno b baia sq mata kanonik, is pleura kai isog nia, giat ta as mmetra sq mata den resan stic m lissec. Ta is pleura tr gwna loip n, kai ta tetr gwna kai ta ex gwna mporo n, an topojethjo n to na d pla sto llo, na qoun tic pleur c touc efapt menec, qwr c ken pou na katastr foun th summetr a. Giat h epif neia pou br sketai g rw ap na shme o kal ptetai ap xi is pleura tr gwna kai xi gwn ec, k je m a ap tic opo ec e nai ta /3 thc orj c, ap t ssera tetr gwna kai t sseric orj c gwn ec, ap tr a ex gwna kai tre c gwn ec exag nou, pou h k je m a touc isoduname me 1 kai 1/3 thc orj c. En tr a pent gwna den arko n gi na kal youn thn epif neia g rw ap na shme o, kai ta t ssera perisse oun. Giat oi tre c gwn ec tou pentag nou isodunamo n me lig tero ap t sseric orj c, en oi t sseric gwn ec isodunamo n me periss tero ap t sseric orj c. En den e nai dunat na qwr soun o te tr a ept gwna me koin c pleur c g rw ap na shme o, giat oi tre c gwn ec eptag nou isodunamo n me 3

4 periss tero ap t sseric orj c. Kai aut isq ei ak mh periss tero gi ta polugwn tera sq mata. Ef son loip n up rqoun tr a sq mata pou na mporo n na kal youn pl rwc thn epif neia g rw ap na shme o, to tr gwno, to tetr gwno kai to ex gwno, oi m lissec e qan th sof a na epil xoun gi to skop touc to sq ma me tic periss terec gwn ec, giat kat laban oti eke no qwr ei periss tero m li ap ta lla d o. Kai oi m lissec b baia x roun m no,ti touc e nai qr simo, oti dhlad to ex gwno e nai megal tero ap to tetr gwno kai to tr gwno kai mpore na qwr sei periss tero m li, en gi thn kataskeu kai twn tri n sqhm twn katanal sketai h dia akrib c pos thta uliko. Eme c mwc pou uposthr zoume oti e maste sof teroi ap tic m lissec, ja exet soume to j ma baj tera. 4

5 EISAGWGH Sthn ergas a aut melet me to klasik isoperimetrik pr blhma ston R n, kaj c kai merik ap ta pio shmantik isoperimetrik probl mata se metriko c q rouc pijan thtac: 1. To isoperimetrik pr blhma sto ep pedo e nai na pr blhma elaq stou: Ap la ta ep peda qwr a pou qoun to dio embad n A, na brejo n eke na pou qoun el qisth per metro. To pr blhma aut apasq lhse touc gewm trec ap thn arqai thta: m sa ap aplo c gewmetriko c sullogismo c, o hn dwroc kat lhxe sthn arq oti o d skoc e nai h l sh tou. Pol arg tera, o Steiner dwse tic pr tec apode xeic tou oti, an up rqei l sh sto pr blhma t te upoqreo tai na e nai o d skoc. H austhr ap deixh thc parxhc l shc e nai sun peia thc isoperimetrik c anis thtac P (K) 4A(K); pou isq ei gia k je qwr o me s noro le a apl kleist kamp lh. Sto pr to m roc aut c thc ergas ac perigr foume nan ap touc sullogismo c tou Steiner, kaj c kai mia ap deixh thc isoperimetrik c anis thtac pou ofe letai ston Hurwitz kai k nei qr sh seir n Fourier. Sto de tero m roc thc ergas ac, d noume d o apode xeic thc isoperimetrik c anis njd n j n 1 n 1 jkj n gia kurt s mata ston R n. H pr th bas zetai sthn anis thta Brunn - Minkowski pou afor thn sq sh an mesa ston gko kai to jroisma Minkowski: An A; B e nai mh ken sumpag upos nola tou R n, t te ja + Bj 1 n jaj 1 n + jbj 1 n : D noume d o apode xeic thc anis thtac Brunn-Minkowski: h m a qrhsimopoie ta leg mena stoiqei dh s nola (Lyusternik), h llh bas zetai sthn apeik nish tou Knothe kai d nei thn anis thta sthn kurt per ptwsh. Gia thn de terh ap deixh thc isoperimetrik c anis thtac, melet me thn m jodo thc summetrikopo hshc kat Steiner. Me thn m jodo aut, xekin ntac ap tuq n kurt s ma kai ektel ntac diadoqik c summetrikopoi seic odhge tai kane c se s mata osod pote kont se mp la. O gkoc diathre tai se k je b ma, en h epif neia mikra nei. Ena epiqe rhma s gklishc pou bas zetai sto je rhma epilog c 5

6 tou Blaschke de qnei oti h mp la e nai h l sh tou isoperimetriko probl matoc an perioristo me sthn kl sh twn kurt n swm twn. D noume d o ak ma efarmog c thc summetrikopo hshc. Parousi zoume thn l sh pou dwse o Blaschke sto pr blhma twn tess rwn shme wn tou Sylvester, kai m a pr sfath apl ap deixh thc anis thtac Blaschke - Santalo gia to gin meno twn gkwn en c summetriko kurto s matoc kai tou poliko tou.. To genikeum no isoperimetrik pr blhma gia nan metrik q ro pijan thtac (X; d; ) diatup netai wc ex c: Gi k je (0; 1) kai k je " > 0 na brejo n eke na ta upos nola A tou X pou qoun m tro (A) = kai h "-ep ktas touc A " = fx X : d(x; A) "g qei to el qisto dunat m tro. An na upos nolo tou X e nai l sh tou parap nw probl matoc gia k je " > 0, t te l me oti qei el qisth epif neia gia to dosm no m tro. Sta pio fusiologik parade gmata metrik n q rwn pijan thtac, h l sh tou parap nw probl matoc e nai h dia gia k je " > 0 kai emfan zei poll c summetr ec: (a) An X = S n 1 e nai h Eukle deia monadia a sfa ra me thn gewdaisiak metrik kai to anallo wto wc proc tic strof c m tro pijan thtac, t te h l sh tou isoperimetriko probl matoc e nai h d-mp la B(x; r) = fx S n 1 : d(x; x 0 ) rg: D noume ap deixh auto tou isqurismo k nontac qr sh thc sfairik c summetrikopo hshc. (b) An X = f 1; 1g n e nai o q roc tou Cantor me thn fusiologik metrik d(x; y) = 1 n jfi n : x i 6= y i gj kai to fusiologik m tro apar jmhshc (A) = jaj= n, t te p li h l sh tou isoperimetriko probl matoc d netai ap tic d-mp lec. (g) An X = R n e nai o q roc tou Gauss me thn Eukle deia metrik kai to m tro pijan thtac pou qei pukn thta thn () n= exp( jxj =), t te h l sh tou isoperimetriko probl matoc d netai ap touc hmiq rouc pou qoun to dosm no m tro. D noume ap deixh auto tou isqurismo me b sh thn sfairik isoperimetrik anis thta kai thn parat rhsh tou Poincare. Se la ta parap nw parade gmata up rqei m a ak ma par metroc: h di stash n. Sun peia thc l shc tou isoperimetriko probl matoc e nai se kajem a ap tic parap nw peript seic h ak loujh parat rhsh: An A X kai (A) = 1, t te gia k je " > 0 isq ei (A " ) 1 c 1 exp( c " n); pou c 1 ; c > 0 ap lutec stajer c. Gia meg lec diast seic n, aut shma nei oti an epekte noume stw kai l go na upos nolo m trou 1= t te aut pou perisse ei e nai exairetik mikr ap thn poyh tou m trou. Oi sumperifor c dhlad twn uposun lwn tou q rou ap thn poyh thc metrik c kai ap thn poyh tou m trou e nai pol diaforetik c. To fain meno 6

7 aut apokale tai {sugk ntrwsh tou m trou}. D noume apeuje ac apode xeic t toiwn ektim sewn gia la ta parade gmata pou anaf rame. To pleon kthm touc e nai oti parak mptoun tic bari c teqnik c pou apaito ntai gia thn akrib l sh twn ant stoiqwn isoperimetrik n problhm twn, ustero n mwc ap thn poyh thc dia sjhshc me thn nnoia oti h f sh twn epiqeirhm twn pou qrhsimopoio ntai e nai m no me thn {eur terh nnoia} gewmetrik. 7

8 PERIEQOMENA I. H ISOPERIMETRIKH ANISOTHTA STO EPIPEDO 1. To isoperimetrik pr blhma sto ep pedo.. H l sh tou isoperimetriko probl matoc ap ton Steiner. 3. H ap deixh thc isoperimetrik c anis thtac ap ton Hurwitz. II. H ISOPERIMETRIKH ANISOTHTA STON R n 1. Orismo Sumbolism c.. H anis thta Brunn - Minkowski. 3. Summetrikopo hsh kat Steiner. 4. Ap stash Hausdor To Je rhma epilog c tou Blaschke. 5. Ap deixh thc isoperimetrik c anis thtac kai thc anis thtac Brunn - Minkowski me th m jodo thc Steiner summetrikopo hshc. 6. To L mma tou C. Borell Sugk ntrwsh tou gkou. 7. To pr blhma twn tess rwn shme wn tou Sylvester. 8. To polik s ma en c summetriko kurto s matoc H anis thta Blaschke - Santalo. 9. H apeik nish tou Knothe Mi tr th ap deixh thc anis thtac Brunn - Minkowski. III. SUGKENTRWSH TOU METROU OIKOGENEIES LEVY 1. H isoperimetrik anis thta sth sfa ra.. Oikog neiec Levy. 3. Diakrit parade gmata oikogenei n Levy. 4. {Isomorfik } isoperimetrik anis thta ston q ro tou Gauss. 8

9 I. H ISOPERIMETRIKH ANISOTHTA STO EPIPEDO 1. To isoperimetrik pr blhma sto ep pedo. 1.1 To isoperimetrik pr blhma sto ep pedo diatup netai me d o tr pouc: (a) San na pr blhma elaq stou: Ap la ta qwr a pou qoun to dio embad n A, na brejo n eke na pou qoun el qisth per metro. (b) San na pr blhma meg stou: Ap la ta qwr a pou qoun thn dia per metro P, na brejo n eke na pou qoun m gisto embad n. 1. Pr pei pr ta na or soume ti akrib c ennoo me me ton ro qwr o: Ta qwr a pou ja mac apasqol soun e nai aut pou qoun san s nor touc mi apl kleist kamp lh. Me ton ro kamp lh ennoo me thn eik na sto ep pedo mi c suneqo c sun rthshc : [a; b]! R. H kamp lh l getai kleist an (a) = (b), kai apl an gi k je t 1 6= t sto [a; b] isq ei (t 1 ) 6= (t ), me m nh exa resh thn per ptwsh t 1 = a; t = b. Gia eukol a ja jewr soume ak ma oti oi kamp lec mac e nai kat tm mata le ec: an (t) = (x(t); y(t)), jewro me oti up rqei mi diam rish a = t 0 < t 1 < : : : < t n = b tou [a; b] t toia ste oi x(t); y(t) na e nai suneq c paragwg simec se k je (t i ; t i+1 ) kai na up rqoun oi pleurik c touc par gwgoi se k je t i. 1.3 Ta probl mata (a) kai (b) e nai du k me thn ex c nnoia: K je l sh tou probl matoc (a) e nai kai l sh tou probl matoc (b), kai ant strofa. Ap deixh: Estw K mi l sh tou probl matoc (a). Dhlad, an A(K) = A(L) t te P (K) P (L). Jewro me opoiod pote qwr o W me P (K) = P (W ). Br skoume > 0 t toio ste A(K) = A(W ) = A(W ). Pr pei na isq ei P (K) P (W ) = P (W ) = P (K), sunep c 1. Ara, A(K) = A(W ) A(W ), dhlad to K qei m gisto embad n gi th dosm nh per metro. To K e nai kai l sh tou probl matoc (b). Me an logo tr po apodeikn etai oti k je l sh tou probl matoc (b) e nai kai l sh tou probl matoc (a). Ap ed kai p ra loip n den ja k noume di krish an mesa sta d o probl mata kai ja mil me apl gi to isoperimetrik pr blhma. Mi ak ma parat rhsh e nai oti h parxh l sewn gia to pr blhma (a) to pr blhma (b) den e nai profan c. Pr pei kane c na apode xei oti up rqoun qwr a me dosm no embad n kai el qisth per metro (ant stoiqa, me dosm nh per metro kai m gisto embad n). 1.4 To isoperimetrik pr blhma apasq lhse touc gewm trec ap thn arqai thta. D noume ed merik apotel smata gi pol gwna, pou tan gnwst ston hn dwro (oc ai nac p.q.) Je rhma. An mesa se la ta n gwna pou qoun stajer embad n A, to kanonik n gwno qei thn el qisth dunat per metro. 9

10 Ap deixh: Estw K na n gwno pou qei to dosm no embad n A kai thn el qisth dunat per metro. Ja apode xoume pr ta oti lec oi pleur c tou e nai sec. Arke na to de xoume gi d o diadoqik c pleur c tou K. Estw AB; BC diadoqik c pleur c me AB 6= BC. Ja bro me na n o n gwno K 1 pou ja qei to dio embad n me to K all mikr terh per metro. Aut e nai topo. Gi to skop aut antikajisto me thn koruf B tou K me mi n llh B 1 kai krat me tic up loipec. H epilog tou B 1 g netai tsi ste na qei thn dia ap stash ap thn AC me to B all to tr gwno AB 1 C na e nai isoskel c (bl pe sq ma). Ta tr gwna ABC kai AB 1 C qoun to dio embad n, ra kai ta K, K 1. Omwc, nac apl c upologism c de qnei oti P (ABC) > P (AB 1 C), ra kai P (K) > P (K 1 ). To K qei loip n lec tou tic pleur c sec, kai m nei na de xoume oti qei kai lec tou tic gwn ec sec. An n = 3, aut e nai faner. Upoj toume loip n oti n 4 kai up rqoun diadoqik c koruf c A; B; C; D tou K t toiec ste oi gwn ec B kai C na e nai nisec, kai ja katal xoume se topo. To ABCD qei b sh AD kai AB = BC = CD, e nai epom nwc mh eggr yimo afo oi gwn ec B kai C e nai nisec. Mporo me t te na to antikatast soume me na llo eggr yimo tetr pleuro AB 1 C 1 D pou qei to dio embad n kai mikr terh per metro. Me thn antikat stash twn B kai C ap ta B 1 ; C 1 pa rnoume na n gwno K 1 t toio ste A(K 1 ) = A(K) = A all P (K 1 ) < P (K). Aut e nai topo, sunep c to K qei lec tou tic gwn ec sec. Sumpera noume oti to K e nai kanonik. L gw tou du smo an mesa sta probl mata (a) kai (b) qoume: Ap la ta n gwna me per metro P, to kanonik qei to m gisto dunat embad n. Upolog zont c to san sun rthsh thc perim trou P pa rnoume: 1.4. Je rhma. Estw n 3 kai P > 0. Gi k je n gwno K n perim trou P isq ei A(K n ) A(Kn) = P 4n cot ; n pou Kn e nai to kanonik n gwno perim trou P. Amesec sun peiec tou Jewr matoc 1.4. e nai oi ex c: (a) H akolouj a n cot n e nai gn sia a xousa kai te nei sto 1. Sunep c, A(K 3 ) < A(K 4 ) < : : : < A(K n)! P 4 : (b) Kan na pol gwno den mpore na e nai l sh tou geniko isoperimetriko probl matoc. An gi par deigma to pol gwno K qei n koruf c, t te gia to kanonik (n + 1) gwno me thn dia per metro isq ei A(K) < A(K n+1): 10

11 (g) An K pol gwno embado A kai perim trou P, t te 4A < P. Gi opoiond pote d sko isq ei is thta: 4A = P. Ep shc, ta Jewr mata kai 1.4. de qnoun oti so pi kanonik e nai na pol gwno kai so periss terec koruf c qei ( ra so pi kont br sketai se k klo), t so pi kont br sketai sto na e nai l sh tou probl matoc. L getai oti, sundu zontac lec tic prohgo menec parathr seic, o hn dwroc kat lhxe sthn diat pwsh thc ak loujhc isoperimetrik c arq c: O d skoc e nai h l sh tou isoperimetriko probl matoc.. H l sh tou isoperimetriko probl matoc ap ton Steiner. To 1841, o Steiner dwse p nte diaforetik c apode xeic tou ex c jewr matoc:.1 Je rhma. An up rqei na qwr o K 0 tou opo ou to embad n den e nai pot mikr tero ap to embad n en c llou qwr ou K me thn dia per metro, t te to K 0 e nai d skoc. Ap deixh: D noume ed thn pr th ap tic apode xeic tou Steiner, pou bas zetai se aplo c gewmetriko c sullogismo c. Upoj toume oti K 0 e nai na qwr o sto ep pedo me m gisto embad n gi dosm nh per metro. B ma 1: To K 0 pr pei na e nai kurt qwr o, dhlad an A; B e nai d o shme a tou K 0 t te ol klhro to euj grammo tm ma AB pr pei na peri qetai sto K 0. Pr gmati, ac upoj soume oti to K 0 den e nai kurt. T te mporo me na bro me d o shme a A kai B tou K 0 pwc sto sq ma: to upodi sthma CD tou AB br sketai xw ap to K 0. En nontac to K 0 me to grammoskiasm no qwr o, pa rnoume na n o qwr o K 1 t toio ste A(K 1 ) > A(K 0 ) kai P (K 1 ) < P (K 0 ), afo to euj grammo tm ma CD qei mikr tero m koc ap thn kamp lh CD. Me kat llhlh omoiojes a tou K 1 prok ptei na qwr o K 0 1 me embad n A(K 0 1) > A(K 0 ) kai per metro P (K 0 1) = P (K 0 ), topo. B ma : To K 0 e nai kurt, kai xekin ntac ap aut ja fti xoume d o lla pi summetrik qwr a K 1 ; K pou qoun thn dia akra a idi thta me to K 0 : ta K 1 ; K qoun thn dia per metro kai to dio ( ra m gisto) embad n me to K 0. H kataskeu twn K 1 kai K g netai wc ex c: sto s noro tou K 0 pa rnoume d o shme a A; B pou to qwr zoun se d o t xa 1 kai me to dio m koc, to mis dhlad thc perim trou tou K 0. An O e nai to m so tou AB, me 0 1 sumbol zoume to summetrik t xo tou 1 wc proc O, kai me 0 to summetrik tou wc proc O. Jewro me to qwr o K 1 pou qei s noro to 1 [ 0 1, kai to qwr o K pou qei s noro to [ 0. Me aut n thn kataskeu, ta K 1 kai K e nai summetrik wc proc O. Ap ton tr po epilog c twn A kai B, gi j = 1; qoume (:1) P (K j ) = l( j ) + l( 0 j) = l( j ) = P (K 0 ); 11

12 pou me l sumbol zoume to m koc kamp lhc. Afo to K 0 qei m gisto embad n gi th dosm nh per metro, (:) A(K 0 ) A(K j ) ; j = 1; : Omwc, (:3) A(K 1 ) + A(K ) = A(K 0 ); ra A(K 1 ) = A(K ) = A(K 0 ). Dhlad, kaj na ap ta K 1 ; K qei ep shc m gisto embad n. B ma 3: Ta K 1 kai K e nai d skoi. Ja de xoume gi par deigma oti to K 1 e nai d skoc de qnontac oti gi k je shme o C sto s nor tou h gwn a ACB e nai orj. Ac upoj soume oti gi k poio shme o C (aut sto sq ma) h ACB den e nai orj. To K 1 e nai summetrik wc proc O, ra to summetrik D tou C wc proc O an kei sto s noro tou K 1, kai to ACBD e nai parallhl grammo. To K 1 apotele tai ap aut to parallhl grammo kai ta t ssera kampul gramma qwr a 1 ; ; 3 ; 4. Fti qnoume na llo qwr o K 0 1 wc ex c: pr ta kataskeu zoume na orjog nio parallhl grammo A 0 C 0 B 0 D 0 me pleur c A 0 C 0 = AC kai C 0 B 0 = CB, kai kat pin stic t sserec pleur c tou koll me ta 1 ; ; 3 ; 4 pwc sto sq ma. Me aut n thn kataskeu, gi to qwr o K 0 pou prok ptei qoume profan c P 1 (K0 1) = P (K 1 ) en (:4) A(K 0 1) A(K 1 ) = A(A 0 C 0 B 0 D 0 ) A(ACBD) > 0; afo to orjog nio A 0 C 0 B 0 D 0 qei thn dia b sh kai megal tero yoc ap to ACBD. Aut e nai topo afo to K 1 qei m gisto embad n gi th dosm nh per metro. Ara, h ACB e nai orj, ki afo to C tan tuq n, to K 1 e nai d skoc. B ma 4: To K 0 = K 1 = K e nai d skoc. Afo ta K 1 ; K e nai d skoi, ta 1 ; e nai hmik klia. Ara kai to K 0 e nai d skoc (kai beba wc K 0 = K 1 = K ). Etsi oloklhr netai h ap deixh. To Je rhma pou m lic apode xame mac l ei pwc an up rqei l sh gi to isoperimetrik pr blhma sto ep pedo, t te aut e nai upoqrewtik o d skoc. Den exasfal zei mwc thn parxh l shc. Enac tr poc gi na parak myoume aut to emp dio e nai o ex c: Gnwr zoume oti gi ton d sko D isq ei P (D) = 4A(D): E dame oti gi k je pol gwno K isq ei P (K) 4A(K). An de xoume oti gi k je qwr o sto ep pedo isq ei h anis thta P (K) 4A(K); 1

13 t te x roume autom twc oti o d skoc qei m gisto embad n gi dosm nh per metro. Pr gmati, an K e nai opoiod pote qwr o me P (K) = P (D) t te A(K) P (K) 4 = P (D) 4 = A(D): Dhlad, o d skoc e nai l sh tou isoperimetriko probl matoc, kai t ra to epiqe rhma tou Steiner apodeikn ei oti e nai kai h monadik tou l sh. Gi mi pl rh loip n l sh tou probl matoc sto ep pedo arke na de xoume thn: Isoperimetrik Anis thta: Gi k je qwr o K sto ep pedo isq ei P (K) 4A(K); pou P h per metroc kai A to embad n tou K. Sthn ep menh par grafo ja d soume mi ap deixh aut c thc anis thtac, h opo a k nei qr sh seir n Fourier kai ofe letai ston Hurwitz (1901). 3. H ap deixh thc isoperimetrik c anis thtac ap ton Hurwitz. Upoj toume ed gi eukol a oti to qwr o K qei san s nor tou mi le a apl kleist kamp lh : [a; b]! R. Me aut ennoo me oti an (t) = (x(t); y(t)), t te oi x 0 kai y 0 e nai suneqe c kai epipl on (x 0 (t); y 0 (t)) 6= (0; 0) gi k je t, to opo o exasfal zei oti h kamp lh qei se k je shme o efapt meno di nusma to opo o metab lletai me suneq tr po. To m koc thc kamp lhc d netai ap thn (3:1) P = b a p [x 0 (t)] + [y 0 (t)] dt: Ja or soume pr ta mi n a parametrikopo hsh thc kamp lhc : Jewro me thn apeik nish s : [a; b]! [0; P ] me t p (3:) s(t) = [x 0 (u)] + [y 0 (u)] du: a H s e nai suneq c kai gn sia a xousa sun rthsh tou t, sunep c or zetai h ant strof thc s 1 sto [0; P ] kai mporo me na jewr soume thn kamp lh 1 : [0; P ]! R me 1 (s) = (t) pou s = s(t). T te, an x 1 (s) = x(t) kai y 1 (s) = y(t) qoume (3:3) kai (3:4) dx 1 ds = dx dt dt ds = x 0 (t) p [x 0 (t)] + [y 0 (t)] dy 1 ds = dy dt dt ds = y 0 (t) p [x 0 (t)] + [y 0 (t)] : 13

14 Ap tic parap nw sq seic bl poume oti h 1 qei thn idi thta (dx 1 =ds) +(dy 1 =ds) = 1 gi k je t. Eqoume dhlad parametrikopoi sei thn kamp lh wc proc m koc t xou. To embad n tou qwr ou K upolog zetai me thn bo jeia tou Jewr matoc tou Green: Jewro me tic sunart seic Q(x; y) = x kai P (x; y) = y. An me 1 sumbol soume kai thn eik na thc kamp lhc 1 (to s noro dhlad tou K), t te (3:5) 1 P dx + Qdy = kai gi tic sugkekrim nec P kai Q pa rnoume (3:6) A(K) = 1 Epetai oti (3:7) A(K) = 1 P kai me olokl rwsh kat par gontec prok ptei (3:8) A(K) = 1 xdy ydx: dxdy; [x 1 (s)y 0 1(s) y 1 (s)x 0 1(s)]ds P 0 x 1 (s)y 0 1(s)ds: Ja k noume ak ma mi allag metablht c: j toume s = P, op te [0; ] kai an () = 1 (s) = (x (); y ()), qoume (3:9) [x 0 ()] + [y 0 ()] = P gi k je, kai (3:10) A(K) = 0 4 x ()y 0 ()d: Ja apode xoume thn ak loujh isoperimetrik anis thta tou Hurwitz: Je rhma. Estw K qwr o sto ep pedo tou opo ou to s noro e nai mi apl kleist kai le a kamp lh. T te, 4A(K) P (K); is thta isq ei m no an to K e nai d skoc. Ap deixh: Mporo me na upoj soume oti to s noro tou K e nai h eik na mi c kamp lhc : [0; ]! R h opo a ikanopoie tic (3.9) kai (3.10). Oi sunart seic x () kai y () e nai suneqe c ra qoun seir c Fourier, kai epeid e nai kai paragwg simec oi seir c Fourier touc sugkl noun s' aut c: 1X (3:11) x () = a 0 + (a k cos k + b k sin k) k=1 14

15 kai (3:1) y () = c 0 X 1 + (c k cos k + d k sin k) : k=1 Epeid oi x 0 kai y 0 e nai suneqe c, qoun seir c Fourier (3:13) x 0 () kai (3:14) y 0 () H is thta tou Parseval mac d nei (3:15) kai (3:16) 0 0 1X k=1 1X k=1 (kb k cos k ka k sin k) (kd k cos k kc k sin k) : [x 0 ()] d = [y 0 ()] d = Sundu zontac me thn (3.9) pa rnoume 1X k=1 1X k=1 k (a k + b k) k (c k + d k): 1X (3:17) P (K) = k (a k + b k + c k + d k): k=1 Ap thn llh pleur, h (3.10) mac d nei (3:18) A(K) = Afair ntac pa rnoume: 0 x ()y 0 ()d = 1X k=1 k(a k d k b k c k ): 1X (3:19) P 4A = k (a k + b k + c k + d k) k(a k d k b k c k ) k=1 1X = k[(a k d k ) + (b k + c k ) ] + k=1 1X (k k) a k + b k + c k + dk 0: H anis thta loip n isq ei kai m nei na exet soume p te mpore na isq ei is thta. Ap thn (3.19) e nai faner oti gi k pr pei na qoume a k = b k = c k = d k = 0 k= 15

16 (afo k k > 0 an k ). Epipl on, to pr to ap ta d o ajro smata pr pei na mhden zetai ki aut, ra a 1 = d 1 kai b 1 = c 1. Dhlad, x () = a 0 + a 1 cos + b 1 sin kai y () = c 0 b 1 cos + a 1 sin : Enac apl c upologism c de qnei oti (3:0) x () a 0 + y () c 0 = a 1 + b 1; dhlad h kamp lh perigr fei k klo, kai to K e nai d skoc. 16

17 II. H ISOPERIMETRIKH ANISOTHTA STON R n. 1. Orismo Sumbolism c. Ja doul youme ston R n, n. O R n e nai efodiasm noc me to eswterik gin meno h; i to opo o ep gei thn Eukle deia n rma jxj = hx; xi 1. Pol suqn ja qrhsimopoio me mi ( qi p nta thn dia) stajer orjokanonik b sh fe 1 ; : : : ; e n g gi to dosm no eswterik gin meno. Me D n sumbol zoume thn Eukle deia monadia a mp la, kai me S n 1 th monadia a sfa ra fx R n : jxj = 1g. Ja qrhsimopoio me ta gr mmata ; ; ; klp gi monadia a dian smata. An S n 1, me? sumbol zoume ton (n 1) di stato up qwro pou e nai orjog nioc sto :? = fx R n : hx; i = 0g: H probol P : R n!? or zetai ap thn P (x) = x hx; i: Ena kurt s ma K ston R n e nai na mh ken, kurt kai sumpag c upos nolo tou R n me mh ken eswterik. Ja l me oti to kurt s ma K e nai summetrik me k ntro summetr ac to 0 an opoted pote x K qoume kai x K. To jroisma kat Minkowski d o mh ken n uposun lwn A kai B tou R n e nai to s nolo A + B := fa + b : a A; b Bg: E kola el gqoume oti an ta A kai B e nai sumpag (ant stoiqa, kurt ), t te kai to jroism touc A + B e nai sumpag c (ant stoiqa, kurt ). Eidik tera, to jroisma d o kurt n swm twn e nai na kurt s ma. Me ton ro stoiqei dec s nolo anafer maste se m a peperasm nh nwsh orjogwn wn pou qoun tic akm c touc par llhlec pr c touc xonec suntetagm nwn (tic dieuj nseic dhlad twn orjokanonik n dianusm twn e j ) kai qoun x na eswterik. Sumbol zoume me I thn kl sh lwn twn stoiqeiwd n sun lwn. An I e nai na t toio orjog nio, me m kh akm n a 1 ; : : : ; a n > 0, t te or zoume ton gko tou na iso tai me jij = a 1 : : : a n : An J = [ m k=1 I k e nai na stoiqei dec s nolo, t te or zoume fusiologik jjj = mx k=1 Estw t ra A ena mh ken, fragm no upos nolo tou R n. Or zoume ton eswterik gko tou A m sw thc jaj = supfjjj : J A; J Ig; ji k j: 17

18 kai ton exwterik gko tou A m sw thc jaj = inffjjj : A J; J Ig: Ja l me oti to A qei gko, kai ja ton sumbol zoume me jaj, an jaj = jaj. Mpore kane c na de xei oti K je kurt s ma ston R n qei gko. Oi idi thtec tou gkou pou qrhsimopoio me suqn sth sun qeia e nai tele wc fusiologik c: (i) O gkoc param nei anallo wtoc wc proc strof c kai metafor c. (ii) An T e nai nac antistr yimoc grammik c metasqhmatism c tou R n, t te gi k je sumpag c upos nolo tou R n isq ei jt (K)j = jdettjjkj: H epif neia (kat Minkowski) en c mh keno, sumpago c uposun lou A tou R n or zetai wc ex c: ja + "D n j = lim inf : "!0 + " Sthn per ptwsh pou to A e nai na kurt s ma ston R n, to parap nw lim inf e nai rio, kai o orism c aut c thc epif neiac e nai isod namoc me k je llo fusiologik orism pou ja mporo se na d sei kane c.. H anis thta Brunn - Minkowski. H anis thta Brunn - Minkowski sund ei ton gko me thn pr xh thc pr sjeshc kat Minkowski. Ja d soume tre c diaforetik c apode xeic, kajem a ap tic opo ec sqet zetai kai me nan pol shmantik k klo ide n. H pr th ap aut c qrhsimopoie ta stoiqei dh s nola kai ofe letai ston Lyusternik (1940):.1 Anis thta Brunn - Minkowski: Estw A kai B sumpag, mh ken upos nola tou R n. T te, (:1) ja + Bj 1 n jaj 1 n + jbj 1 n : Ap deixh: 1h Per ptwsh: Exet zoume pr ta thn per ptwsh pou ta A kai B e nai orjog nia me tic akm c touc par llhlec proc touc xonec suntetagm nwn. Upoj toume oti a 1 ; : : : ; a n > 0 e nai ta m kh twn akm n tou A, kai b 1 ; : : : ; b n > 0 ta m kh twn akm n tou B. T te, to A + B e nai ki aut orjog nio me tic akm c tou par llhlec proc touc xonec suntetagm nwn kai ant stoiqa m kh a 1 + b 1 ; : : : ; a n + b n. Epom nwc, h (.1) pa rnei s ut n thn per ptwsh th morf (:) ((a 1 + b 1 ) : : : (a n + b n )) 1 n (a1 : : : a n ) 1 n + (b 1 : : : b n ) 1 n : 18

19 Isod nama, zht me na de xoume oti (:3) 1 a1 a n : : : a 1 + b 1 a n + b n n + b1 a 1 + b 1 : : : 1 b n n 1: a n + b n Omwc ap thn anis thta arijmhtiko gewmetriko m sou to arister m loc thc (.3) e nai mikr tero so ap 1 a1 (:4) + : : : + a n + 1 b1 + : : : + b n = 1: n a 1 + b 1 a n + b n n a 1 + b 1 a n + b n Dhlad, h anis thta Brunn - Minkowski isq ei s ut n thn apl per ptwsh. h Per ptwsh: Upoj toume oti ta A; B e nai stoiqei dh s nola, kaj na dhlad ap ut e nai peperasm nh nwsh orjogwn wn pou qoun x na eswterik kai akm c par llhlec proc touc xonec suntetagm nwn. Or zoume san poluplok thta tou zeugario (A; B) to sunolik pl joc twn orjogwn wn pou sqhmat zoun ta A; B. Ja apode xoume thn (.1) me epagwg wc proc thn poluplok thta m tou (A; B). Otan m =, ta A kai B e nai orjog nia kai h (.1) qei dh apodeiqje. Upoj toume loip n oti m 3 kai oti h (.1) isq ei gi zeug ria stoiqeiwd n sun lwn me poluplok thta m 1. Afo m 3, k poio ap ta A kai B ( stw to A) apotele tai ap toul qiston d o orjog nia. Estw I 1, I du ap ut. Ta I 1 kai I qoun x na eswterik, sunep c mporo me na ta diaqwr soume me na uperep pedo par llhlo proc k poion k rio up qwro tou R n. Qwr c bl bh thc genik thtac upoj toume oti aut to uperep pedo perigr fetai ap thn x n = gi k poio R. Or zoume (:5) A + = A \ fx R n : x n g ; A = A \ fx R n : x n g: Ta A + kai A e nai stoiqei dh s nola, qoun x na eswterik kai kaj na touc sqhmat zetai ap lig tera orjog nia ap ti to A (To uperep pedo x n = to pol pol na k bei k je orjog nio tou A se du orjog nia, na sto A + ki na sto A. Omwc, to I 1 peri qetai ex olokl rou sto A + en to I sto A ). Pern ntac t ra sto B, br skoume uperep pedo x n = s t toio ste an B + = B \ fx R n : x n sg kai B = B \ fx R n : x n sg na isq ei (:6) ja + j jaj = jb+ j jbj : Ta B + kai B e nai stoiqei dh s nola me pl joc orjogwn wn pou den xepern ei aut tou B. Onom zoume ton koin l go gkwn sthn (.6). Profan c, 0 < < 1. E nai faner oti (:7) A + B = (A + + B + ) [ (A + + B ) [ (A + B + ) [ (A + B ): Ap thn llh pleur, afo A + + B + fx : x n + sg kai A + B fx : x n + sg, ta A + + B + kai A + B qoun x na eswterik. Qrhsimopoi ntac kai thn (.7) pa rnoume (:8) ja + Bj ja + + B + j + ja + B j: 19

20 Me b sh thn kataskeu pou k name, efarm zetai h epagwgik up jesh sto dexi m loc: (:9) ja + + B + j 1 n ja+ j 1 n + jb + j 1 n ; ja + B j 1 n ja j 1 n + jb j 1 n ; op te k nontac pr xeic sthn (.8) kai pa rnontac up yh thn (.6) qoume ja + Bj (jaj) 1=n + (jbj) 1=n n + ((1 )jaj) 1=n + ((1 )jbj) 1=n n = [jaj 1=n + jbj 1=n ] n + (1 )[jaj 1=n + jbj 1=n ] n = [jaj 1=n + jbj 1=n ] n ; apopou petai oti (:10) ja + Bj 1 n jaj 1 n + jbj 1 n : Genik Per ptwsh: Estw A; B tuq nta mh ken sumpag upos nola tou R n. Up rqoun akolouj ec fa m g kai fb m g stoiqeiwd n sun lwn me tic idi thtec A A m ; ja m j! jaj ; B B m ; jb m j! jbj ; m N: T te A + B A m + B m gi k je m N kai mporo me na epil xoume ta A m ; B m tsi ste na qoume kai ja m + B m j! ja + Bj, ra (:11) ja + Bj 1 n = lim m!1 ja m + B m j 1 n lim m!1 [ja mj 1 n + jb m j 1 n ] = jaj 1 n + jbj 1 n : H anis thta Brunn - Minkowski gi kurt s mata ston R n suqn diatup netai kai wc ex c:. P risma. Estw K 1 ; K kurt s mata ston R n. Gi k je (0; 1) isq ei (:1) jk 1 + (1 )K j 1=n jk 1 j 1=n + (1 )jk j 1=n : Pi isqur, h sun rthsh f : [0; 1]! R me f() = jk 1 +(1 )K j 1=n e nai ko lh. Ap deixh: Arke na de xoume oti f(a + (1 a)) af() + (1 a)f() gi k je ; [0; 1] kai a (0; 1). Eqoume (:13) f(a + (1 a)) = j(a + (1 a))k 1 + (1 a (1 a))k j 1 n = j(a + (1 a))k 1 + (a(1 ) + (1 a)(1 ))K j 1 n = ja(k 1 + (1 )K ) + (1 a)(k 1 + (1 )K )j 1 n ja(k 1 + (1 )K )j 1 n + j(1 a)(k 1 + (1 )K )j 1 n = ajk 1 + (1 )K j 1 n + (1 a)jk 1 + (1 )K j 1 n = af() + (1 a)f(); pou qrhsimopoi same to gegon c oti an X R n kurt, mh ken kai a; b > 0 t te ax + bx = (a + b)x. 0

21 Mi llh sun peia thc anis thtac Brunn - Minkowski e nai h ak loujh anis thta (h opo a e nai anex rthth thc di stashc):.3 P risma. Estw A; B sumpag, mh ken upos nola tou R n. Gi k je (0; 1) qoume (:14) ja + (1 )Bj jaj jbj 1 : Ap deixh: H sun rthsh log e nai ko lh, ki aut qei san sun peia thn (:15) x y 1 x + (1 )y gi k je x; y > 0 kai (0; 1). Ap thn anis thta Brunn - Minkowski pa rnoume ja+(1 )Bj [jaj 1=n +(1 )jbj 1=n ] n [jaj =n jbj (1 )=n ] n = jaj jbj 1 : Mi pr th shmantik sun peia thc anis thtac Brunn - Minkowski e nai h isoperimetrik anis thta gi kurt s mata ston R n :.4 Je rhma. Estw K kurt s ma ston R n me jkj = jd n j. T n ): Ap deixh: Me b sh ton orism thc epif neiac, jk + "D n j jkj [jkj = lim 1=n + j"d n j 1=n ] n jkj lim "!0 + " "!0 + " = [jd nj 1=n + j"d n j 1=n ] n jd n j " jd n j[(1 + ") n 1] = lim = njd n j: "!0 + " Omwc, o ant stoiqoc upologism c gia n ) d n ) = njd n j (h anis thta Brunn - Minkowski isq ei ed san is thta). n ): Ara, h mp la e nai l sh tou isoperimetriko probl matoc: an mesa se la ta kurt s mata tou R n pou qoun dosm no gko V, h mp la qei thn mikr terh epif neia. Mi llh sun peia thc anis thtac Brunn - Minkowski e nai to Je rhma tou Brunn gi tic (n 1) di statec tom c en c kurto s matoc. Pi sugkekrim na, jewro me na kurt s ma K ston R n, na monadia o di nusma S n 1 kai thn sun rthsh f : R! R me (:18) f(t) = jk \ (? + t)j 1

22 pou? e nai o (n 1) di statoc up qwroc o k jetoc sto. H sun rthsh f metr ei to {embad n} twn par llhlwn tom n tou K pou e nai k jetec sto. Gi thn f isq ei to ex c:.5 Je rhma. Estw K kurt s ma ston R n kai S n 1. T te, h sun rthsh g(t) = [f(t)] 1 n 1 = jk \ (? + t)j 1 n 1 e nai ko lh ston for a thc. Ep shc, h log f e nai ko lh ston for a thc. Ap deixh: Taut zoume ton? me ton R n 1 kai gr foume K(t) gi thn probol thc tom c K \ (? + t) ston?. Dhlad, (:19) K(t) = fx R n 1 : (x; t) Kg: Ja apode xoume oti gi k je t; s R kai (0; 1) (:0) K(t) + (1 )K(s) K(t + (1 )s): Pr gmati, stw x K(t) kai y K(s). T te, (x; t) K kai (y; s) K kai afo to K e nai kurt pa rnoume (x + (1 )y; t + (1 )s) K, dhlad x + (1 )y K(t + (1 )s). Afo jk(t)j = jk \ (? + t)j gi k je t R, to gegon c oti h g e nai ko lh prok ptei mesa ap thn anis thta Brunn - Minkowski. Sunj tontac me thn ko lh sun rthsh log bl poume oti h log g ra kai h log f e nai ko lh ston for a thc. Poll c diaisjhtik faner c prot seic pou aforo n gkouc kurt n swm twn apodeikn ontai me thn bo jeia thc anis thtac Brunn - Minkowski. H qr sh thc e nai sun jwc ousiastik, me thn nnoia oti e nai d skolo an qi ad nato na doje pi stoiqei dhc ap deixh. Kle noume aut n thn par grafo me d o t toia parade gmata:.6 P risma. Estw K na summetrik kurt s ma ston R n, kai S n 1. T te, (:1) max tr jk \ (? + t)j = jk \? j: Dhlad, h m gisth tom tou K k jeta sto e nai h kentrik : aut pou pern ei ap to k ntro summetr ac tou K. Ap deixh: Amesh ap to Je rhma.5. H g e nai t ra rtia kai ko lh, sunep c pa rnei th m gisth tim thc sto 0..7 P risma. Estw A; B kurt s mata ston R n, kai oti ta A; B e nai summetrik, me k ntro summetr ac to 0. T te, gi k je y R n qoume (:) ja \ (y + B)j ja \ Bj: Ap deixh: Ja de xoume oti gi k je y R n (:3) 1 [A \ (y + B)] + 1 [A \ ( y + B)] A \ B:

23 Pr gmati: an z A \ (y + B) kai w A \ ( y + B), t te (z + w)= A l gw kurt thtac tou A. Ep shc, up rqoun b 1 ; b B t toia ste z = y + b 1 kai w = y + b, op te (z + w)= = (b 1 + b )= B. Ara, h (.3) isq ei. Ap to P risma.3 pa rnoume (:4) ja \ Bj ja \ (y + B)j 1 ja \ ( y + B)j 1 : Omwc, qrhsimopoi ntac thn summetr a twn A kai B mporo me na do me oti (:5) [A \ (y + B)] = A \ ( y + B): Giat z A \ ( y + B) an kai m no an z A kai up rqei b B t toio ste z = y + b, dhlad an kai m no an z A kai z = y + ( b) y + B. Ta A \ (y + B) kai A \ ( y + B) qoun epom nwc ton dio gko, kai h (.4) mac d nei (:6) ja \ Bj ja \ (y + B)j: 3. Summetrikopo hsh kat Steiner. Sthn par grafo aut ja perigr youme mi diadikas a me thn opo a xekin ntac ap tuq n kurt s ma K odhgo maste me {omal tr po} se lo kai pi summetrik kurt s mata. H diadikas a aut qrhsimopoie tai pol suqn tan j loume na de xoume oti h mp la e nai l sh en c probl matoc meg stou elaq stou. Estw K na kurt s ma ston R n. Wc sun jwc, me sumbol zoume na monadia o di nusma (kate junsh) kai me? ton (n 1) di stato up qwro pou e nai k jetoc sto : (3:1)? = fx R n : hx; i = 0g: H probol P (K) tou K sthn die junsh tou apotele tai ap lec tic probol c shme wn tou K ston?. E nai e kolo na de kane c oti (3:) P (K) = fx? : 9t R : x + t Kg: 3.1 L mma. Gi k je kurt s ma K kai k je S n 1, to P (K) e nai kurt s ma. Ap deixh: De qnoume pr ta oti to P (K) e nai kurt. Estw x; y P (K) kai (0; 1). Up rqoun pragmatiko arijmo t kai s t toioi ste x + t K kai y + s K. Afo to K e nai kurt, qoume (3:3) (x + t) + (1 )(y + s) = [x + (1 )y] + [t + (1 )s] K; op te h (3.) mac d nei oti x + (1 )y P (K). Ara, to P (K) e nai kurt. Ja de xoume oti to P (K) e nai kleist : Estw fx n g mi akolouj a shme wn tou P (K) kai ac upoj soume oti x n! x. Up rqei mi akolouj a pragmatik n arijm n 3

24 ft n g t toia ste x n + t n K. Epeid to K e nai fragm no, h ft n g e nai fragm nh ra qei sugkl nousa upakolouj a ft kn g. An t kn! t, qoume x kn + t kn! x + t, kai afo to K e nai kleist petai oti x + t K. Sunep c x P (K) kai to P (K) e nai kleist. To oti to P (K) e nai fragm no kai qei mh ken eswterik e nai profan c: to K peri qetai se mi mp la me k ntro to 0, ra to P (K) ja peri qetai se mp la tou? thc diac akt nac. Ep shc, to K e nai s ma ra peri qei mp la k poiac jetik c akt nac. To dio ja sumba nei me opoiad pote (n 1) di stath probol tou (h probol mp lac e nai mp la me thn dia akt na). Gi na or soume thn Steiner summetrikopo hsh tou K sthn kate junsh tou, ja qreiaste na gr youme to K se mi pi bolik morf : 3. L mma. Estw K kurt s ma ston R n kai S n 1. Or zoume d o sunart seic f; g : P (K)! R me (3:4) f(x) = minft R : x + t Kg ; g(x) = maxft R : x + t Kg: T te, h f e nai kurt kai h g e nai ko lh. Ap deixh: Ja de xoume oti h f e nai kurt. Arke na de xoume oti gi k je x; y P (K) isq ei f( x+y f (x)+f (y) ). Ac upoj soume oti f(x) = t kai f(y) = s. T te, x + t K kai y + s K. Epeid to K e nai kurt, pa rnoume (3:5) (x + t) + (y + s) Ap ton orism thc f petai oti x + y (3:6) f t + s = x + y = + t + s K: f(x) + f(y) : Ara h f e nai kurt, kai me ton dio tr po de qnoume oti h g e nai ko lh. S mfwna me to L mma 3. mporo me na gr youme to s ma K sth morf (3:7) K = fx + t : x P (K) ; f(x) t g(x)g: Ant strofa, an na qwr o K ston R n qei san probol ston? na kurt s ma P (K) kai gr fetai sth morf (3.7), pou f kurt, g ko lh kai f g sto P (K), t te to K e nai kurt. Or zoume t ra thn Steiner summetrikopo hsh tou K sthn kate junsh tou wc ex c: (3:8) S (K) = fx + t : x P (K) ; jtj g(x) f(x) g: Dhlad, gi k je x P (K) jewro me na euj grammo tm ma par llhlo sto me m koc g(x) f(x) kai m so to x, kai pa rnoume thn nwsh lwn aut n twn eujugr mmwn tmhm twn. 4

25 E nai faner oti to S (K) e nai summetrik wc proc? (aut h parat rhsh dikaiologe ton ro {summetrikopo hsh} ). Oi basik c idi thtec tou metasqhmatismo S perigr fontai sto ep meno l mma: 3.3 L mma. Estw K kurt s ma ston R n kai S n 1. T te, to S (K) e nai kurt s ma, kai js (K)j = jkj. Ap deixh: Estw x + t; y + s S (K) kai (0; 1). T te, x; y P (K) kai afo to P (K) e nai kurt qoume x + (1 )y P (K). Ap thn llh pleur, g(x) f(x) g(y) f(y) jt + (1 )sj jtj + (1 )jsj + (1 ) (3:9) 1 [g(x + (1 )y) f(x + (1 )y)] epeid h g f e nai ko lh. Sunep c, (x + t) + (1 )(y + s) S (K) kai to S (K) e nai kurt. Gi ton de tero isqurism parathro me oti (g f)(x) (g f)(x) (3:10) js (K)j = ( ) dx = P (K) P (K) [g(x) f(x)]dx = jkj: Ja de xoume oti me k je summetrikopo hsh Steiner h epif neia opoioud pote kurto s matoc mikra nei. Gi to skop aut qreiaz maste na ak ma l mma: 3.4 L mma. Estw K; L kurt s mata ston R n kai S n 1. T te, (3:11) S (K) + S (L) S (K + L): Ap deixh: X roume oti (3:1) S (K) = fx + t : x P (K); jtj (g K f K )(x) g kai (3:13) S (L) = fy + s : y P (L); jsj (g L f L )(y) g: Ara, (3:14) S (K) + S (L) = fx + y + (t + s) : x P (K); y P (L); jtj (g K f K )(x) ; jsj (g L f L )(y) g: 5

26 Ap thn llh pleur, (3:15) S (K + L) = fz + : z P (K + L); jj (g K+L f K+L )(z) g: Estw x + t S (K) kai y + s S (L). T te, x P (K) kai y P (L) ra up rqoun t 1 ; s 1 R t toia ste x + t 1 K kai y + s 1 L. Epetai ti x + y + (t 1 + s 1 ) K + L, sunep c x + y P (K + L). Ep shc, (3:16) jt + sj jtj + jsj g K(x) f K (x) + g L(y) f L (y) = (g K(x) + g L (y)) (f K (x) + f L (y)) op te s mfwna me tic (3.14) kai (3.15) gi thn (3.11) arke na de xoume oti gi k je x; y P (K) (3:17) g K (x) + g L (y) g K+L (x + y) ; f K (x) + f L (y) f K+L (x + y): Ja de xoume thn pr th ap tic d o anis thtec: Eqoume x + g K (x) K kai y + g L (y) L, epom nwc x + y + (g K (x) + g L (y)) K + L apopou sumpera noume oti g K (x) + g L (y) g K+L (x + y). 3.5 Je rhma. Estw K kurt s ma ston R n kai S n 1. T te, Ap deixh: Estw " > 0. E kola bl poume oti S ("D n ) = "D n kai to L mma 3.4 mac exasfal zei oti (3:19) S (K) + "D n = S (K) + S ("D n ) S (K + "D n ): H Steiner summetrikopo hsh diathre touc gkouc, sunep c js (K)j = jkj kai js (K + "D n )j = jk + "D n j (L mma 3.3). Epetai oti (3:0) js (K) + "D n j js (K)j js (K + "D n )j js (K)j = jk + "D nj jkj : " " " Pa rnontac ria kaj c "! 0 + bl poume Or zoume thn di metro diam(k) tou kurto s matoc K j tontac (3:1) diam(k) = maxfjz 1 z j : z 1 ; z Kg: H ep menh pr tash de qnei oti me Steiner summetrikopo hsh opoioud pote kurto s matoc wc proc opoiad pote kate junsh, h di metroc mikra nei: 6

27 3.6 Pr tash. Estw K kurt s ma ston R n kai S n 1. T te, (3:) diam(s (K)) diam(k): Ap deixh: Estw z 1 ; z S (K) t toia ste jz 1 z j = diam(s (K)). E kola bl poume oti ta z 1 ; z pr pei na e nai thc morf c (3:3) z 1 = x 1 + g(x 1) f(x 1 ) ; z = x g(x ) f(x ) gi k poia x 1 ; x P (K). Gr foume g i = g(x i ) kai f i = f(x i ), kai jewro me ta shme a (3:4) w i = x i + g i ; v i = x i + f i ; i = 1; : Ta t ssera aut shme a an koun sto K, kai qrhsimopoi ntac to Pujag reio Je rhma kai apl c pr xeic bl poume oti (3:5) jw 1 v j +jw v 1 j jz 1 z j = jg 1 f j +jg f 1 j j g 1 f 1 + g f j 0: Omwc, jw 1 v j; jw v 1 j diam(k) op te ap thn (3.5) e nai faner oti (3:6) diam(s (K)) = jz 1 z j maxfjw 1 v j; jw v 1 jg diam(k): 4. Ap stash Hausdor To Je rhma epilog c tou Blaschke. 4.1 Orism c. Estw K; L mh ken, sumpag upos nola tou R n. Or zoume thn ap stash Hausdor twn K kai L wc ex c: (4:1) d(k; L) = minf 0 : K L + D n ; L K + D n g: E nai e kolo na do me oti h d e nai metrik : Gi k je K; L isq ei d(k; L) 0 ex rismo, kai an d(k; L) = 0 t te K L + 0D n = L + f0g = L kai moia L K. Dhlad, K = L. H (4.1) e nai profan c summetrik wc proc K kai L, sunep c d(k; L) = d(l; K). Gi k je K; L kai M qoume d(k; L) d(k; M)+d(M; L): Estw a = d(k; M) kai b = d(m; L). T te, (4:) K M + ad n ; M K + ad n ; M L + bd n ; L M + bd n : Epetai oti kai K M + ad n L + bd n + ad n = L + (b + a)d n L M + bd n K + ad n + bd n = K + (a + b)d n ; 7

28 dhlad d(k; L) a + b. [Qrhsimopoi same p li to gegon c oti an X kurt kai a; b > 0, t te ax + bx = (a + b)x.] 4. Orism c. Estw K mh ken, sumpag c upos nolo tou R n. H kurt j kh co(k) tou K e nai to mikr tero kurt upos nolo tou R n pou peri qei to K. Mpore kane c e kola na de oti (4:3) co(k) = f 1 x 1 + : : : + m x m : m N; x i K; i 0; mx i=1 i = 1g: Mi apl parat rhsh e nai oti to K e nai kurt an kai m no an co(k) = K. To l mma pou akolouje de qnei oti pa rnontac kurt c j kec mei noume thn ap stash Hausdor: 4.3 L mma: Estw K; L mh ken, sumpag upos nola tou R n. T te, (4:4) d(k; L) d(co(k); co(l)): Ap deixh: J toume a = d(k; L). T te, K L + ad n kai L K + ad n. Estw x co(k). Ap thn (4.3) up rqoun x 1 ; : : : ; x m K kai 1 ; : : : ; m 0 me P i = 1 t toia ste x = 1 x 1 + : : : + m x m. Gi k je i = 1; : : : ; m mporo me na bro me y i L t toio ste jx i y i j a (epeid K L + ad n ). T te, y = 1 y 1 + : : : + m y m co(l) kai jx yj mx i=1 i jx i y i j = a mx i=1 i = a: Epetai oti co(k) co(l)+ad n. Me an logo tr po bl poume oti co(l) co(k)+ ad n, dhlad d(co(k); co(l)) a = d(k; L). 4.4 Par deigma. Jewro me san K nan d sko akt nac R > 0 kai san L thn perif rei tou. T te, co(k) = K = co(l) ra d(co(k); co(l)) = 0 en d(k; L) = R. 4.5 Orism c. Estw fk m g mn akolouj a mh ken n, sumpag n uposun lwn tou R n. L me oti h fk m g sugkl nei se k poio sumpag c K R n kai gr foume K m! K an d(k m ; K)! Pr tash. Estw fk m g akolouj a mh ken n, kurt n kai sumpag n uposun lwn tou R n kai K R n sumpag c t toio ste K m! K. T te, to K e nai kurt. Ap deixh: Estw " > 0. Up rqei m N t toioc ste d(k m ; K) < ". Omwc, to K m e nai kurt ra K m = co(k m ) kai ap to L mma 4.3 bl poume oti d(k m ; co(k)) = d(co(k m ); co(k)) d(k m ; K) < ": H trigwnik anis thta gi thn d d nei d(k; co(k)) d(k m ; K) + d(k m ; co(k)) < "; 8

29 kai afo to " tan tuq n sumpera noume oti d(k; co(k)) = 0 dhlad K = co(k). Epetai oti to K e nai kurt. To basik apot lesma aut c thc paragr fou e nai to Je rhma Epilog c tou Blaschke s mfwna me to opo o ta sumpag kurt upos nola opoiasd pote mp lac sqhmat zoun nan sumpag metrik q ro me thn metrik Hausdor: 4.7 Je rhma. Estw R > 0 kai fk m g mn mi akolouj a mh ken n, sumpag n kurt n uposun lwn thc RD n. T te, up rqoun upakolouj a fk lm g thc fk m g kai kurt sumpag c K RD n t toia ste K lm! K. Gi thn ap deixh ja qreiasto me to ak loujo l mma: 4.8 L mma. Estw " > 0. Me tic upoj seic tou Jewr matoc, mporo me na epil xoume upakolouj a deikt n l 1 < l < : : : < l m < : : : me thn idi thta gi k je i; j N. d(k li ; K lj ) " Ap deixh: H mp la RD n e nai sumpag c, epom nwc gi to dosm no " > 0 mporo me na bro me x 1 ; : : : ; x N RD n t toia ste (4:5) RD n N[ i=1 (x i + " D n): Or zoume E = fx 1 ; : : : ; x N g. Gi k je m N j toume E m = E\(K m + " D n). K je E m e nai peperasm no s nolo, ra sumpag c. Ja de xoume oti d(k m ; E m ) ". Ap ton orism tou E m e nai faner oti E m K m + " D n. Estw x K m. T te x RD n, sunep c up rqei i N t toio ste jx x i j ". Aut shma nei oti x i x + " D n K m + " D n, ra x i E m. Ap thn llh pleur, x x i + " D n E m + " D n. Ara K m E m + " D n, kai d(k m ; E m ) ". Gi k je m N qoume bre E m E me thn idi thta d(k m ; E m ) ". Omwc to E e nai peperasm no s nolo, ra up rqoun peperasm nec (lig terec ap N ) epilog c uposun lwn tou. Epetai oti up rqoun peiroi to pl joc de ktec l 1 < l < : : : < l m < : : : kai monadik E E ste d(k lj ; E ) ". An t ra i 6= j, h trigwnik anis thta d nei d(k li ; K lj ) d(k li ; E ) + d(e ; K lj ) " + " = ": Ap deixh tou Jewr matoc: Efarm zoume diadoqik to L mma 4.8 me " = 1 m : Gi " = 1 mporo me na bro me upakolouj a K 11 ; K 1 ; : : : ; K 1m ; : : : thc fk m g t toia ste an i 6= j na isq ei d(k 1i ; K 1j ) 1: Pa rnoume t ra " = 1 kai br skoume upakolouj a K 1; K ; : : : ; K m ; : : : thc fk 1j g t toia ste an i 6= j na isq ei d(k i ; K j ) 1 : 9

30 Suneq zontac tsi or zoume diadoqik upakolouj a thc upakolouj ac tsi ste gia thn m ost upakolouj a fk mj g na isq ei d(k mi ; K mj ) 1 m gi k je i 6= j. Jewro me t ra thn diag nia upakolouj a K m = K mm. Aut e nai upakolouj a thc fk m g kai ikanopoie to ex c: An l; s N kai l < s, t te oi K l ; K s e nai roi thc fk lj g, sunep c (4:6) d(k l ; K s ) 1 l : Or zoume to s nolo (4:7) K = 1\ l=1 (K l + 1 l D n); kai ja de xoume oti Km! K. Akrib stera, ja de xoume oti d(km; K) m 1 gi k je m N. Ap ton orism tou K e nai faner oti K Km + m 1 D n. Estw x Km. Ap thn (4.6), gi k je s > m mporo me na bro me x s Ks t toio ste jx x s j m 1. Eqoume x s RD n gi k je s, ra up rqoun upakolouj a fx rs g thc fx s g kai y RD n t toia ste x rs! y. Afo jx x rs j 1=m, e nai faner oti (4:8) jx yj 1 m : Ja de xoume oti y K. Pr pei dhlad na exasfal soume oti y K l + 1 l D n gi k je l N. Omwc, telik qoume r s > l kai ap thn (4.6) bl poume oti an r s > l t te (4:9) x rs K r s K l + 1 l D n: Pern ntac sto rio, sumpera noume oti y K l + 1 l D n, kai epeid to l tan tuq n qoume y K. H (4.8) t ra mac d nei (4:10) x y + 1 m D n K + 1 m D n; kai epeid to x K m tan tuq n, K m K + 1 m D n. Ara, d(k m; K) 1 m, dhlad K m! K. To ti to K e nai sumpag c kai kurt petai ap thn (4.7). To ti to K e nai mh ken petai ap thn ap deixh (bl pe (4.10}. 30

31 5. De terh ap deixh thc isoperimetrik c anis thtac kai thc anis thtac Brunn - Minkowski me th m jodo thc Steiner summetrikopo hshc. O pr toc mac st qoc s ut n thn par grafo e nai na de xoume oti xekin ntac ap opoiod pote kurt s ma ston R n mporo me me peperasm nec to pl joc diadoqik c Steiner summetrikopoi seic na to f roume osod pote kont se mi mp la. K nontac qr sh auto tou apotel smatoc ja d soume mi de terh ap deixh t so thc isoperimetrik c anis thtac so kai thc anis thtac Brunn - Minkowski. 5.1 L mma. Estw fk m g mi akolouj a kurt n swm twn pou sugkl nei (me thn Hausdor metrik ) se na kurt s ma K me r 1 D n K s 1 D n, 0 < r 1 < s 1. T te, gi k je S n 1, (5:1) S (K m )! S (K): Ap deixh: Mporo me na upoj soume oti gi k poia 0 < r < r 1 kai s > s 1 kai gi k je m m 1 isq ei rd n K m sd n. Estw " > 0. J loume na bro me m 0 N me thn idi thta d(s (K m ); S (K)) < ", dhlad (5:) S (K m ) S (K) + "D n ; S (K) S (K m ) + "D n gi k je m m 0. Pa rnoume " 0 = r s " kai ap thn up jesh tou L mmatoc mporo me na bro me m N t toio ste (5:3) K m K + r s "D n ; K K m + r s "D n gi k je m m. Epil goume m 0 = maxfm 1 ; m g. An m m 0, t te (5:4) K m K + r s "D n K + " s K = 1 + " K: s Epetai oti (5:5) S (K m ) 1 + " s Me ton dio tr po de qnoume oti S (K) = S (K)+"S ( 1 s K) S (K)+"S (D n ) = S (K)+"D n : (5:6) S (K) S (K m ) + "D n : Ara, d(s (K); S (K m )) < " gi k je m m 0. Dhlad, S (K m )! S (K). 5. Je rhma. Estw K kurt s ma ston R n, pou peri qei to 0 sto eswterik tou. Jewro me thn kl sh lwn twn peperasm nwn Steiner summetrikopoi sewn tou K (5:7) S(K) = fs l : : : S 1 (K) : i S n 1 ; l Ng: T te, gi k je " > 0 mporo me na bro me C S(K) t toio ste d(c; D n ) < ", pou > 0 h akt na gi thn opo a jkj = jd n j. 31

32 Ap deixh: Jewro me ton arijm (5:8) = inffr > 0 : 9C S(K) : C rd n g: Gi k je " > 0 mporo me na bro me C " t toio ste C " ( + ")D n. Pa rnontac " = 1=m; m N, br skoume mi akolouj a fc m g kurt n swm twn me C m S(K) kai C m ( + 1 m )D n. Ola ta s mata C m peri qontai sthn ( + 1)D n, epom nwc efarm zetai to Je rhma Epilog c tou Blaschke: up rqei mi upakolouj a fc km g thc fc m g me thn idi thta C km! C, gi k poio sumpag c kurt s nolo C. Ja de xoume oti C = D n. E nai e kolo na do me oti C D n : An " > 0, t te gi meg la m qoume (5:9) C C km + "D n ( + 1 k m )D n + "D n = ( + 1 k m + ")D n : Afo 1=k m! 0 kai to " tan tuq n, C D n. Ac upoj soume oti to C e nai gn sio upos nolo thc D n. Up rqei dhlad x 0 S n 1 me x 0 = C. To C e nai kleist, epom nwc up rqei > 0 t toio ste [x 0 + D n ] \ C = ;. Mporo me na bro me peperasm na to pl joc shme a x 1 ; : : : ; x N S n 1 t toia ste (5:10) S n 1 N[ i=0 [x i + D n ] \ S n 1 : Jewro me monadia o di nusma 1 sthn die junsh tou euj grammou tm matoc [x 0 x 1 ], kai summetrikopoio me to C sthn kate junsh tou 1. Mporo me t te na do me oti (5:11) 1[ i=0 [x i + D n ] \ S n 1! \ S 1 C = ;: Suneq zoume summetrikopoi ntac to S 1 C sthn kate junsh en c monadia ou dian smatoc par llhlou proc to euj grammo tm ma [x 0 x ]. Opwc prin, (5:1) [ i=0 [x i + D n ] \ S n 1! \ S S 1 C = ;: Met ap N b mata, katal goume sto kurt s ma C 1 l gw thc (5.10) qei thn idi thta = S N : : : S 1 C, to opo o (5:13) C 1 \ (S n 1 ) = ;: L gw sump geiac sumpera noume oti up rqei " > 0 t toio ste (5:14) C 1 ( ")D n : 3

33 Ap thn llh pleur, afo C km! C me diadoqik c efarmog c tou L mmatoc 5.1 bl poume oti (5:15) S N : : : S 1 C km! C 1 : Aut shma nei oti up rqei m N arket meg lo t toio ste (5:16) S N : : : S 1 C km C 1 + " D n ( " )D n: Aut e nai topo, giat C km S(K) ra kai S N : : : S 1 C km S(K), to opo o antif skei proc thn (5.8). Ara, C = D n kai afo C km! D n gi k je " > 0 up rqei C S(K) t toio ste d(c; D n ) < ". To oti jd n j = jkj e nai apl sun peia tou oti h Steiner summetrikopo hsh diathre touc gkouc: k je C S(K) qei gko jcj = jkj, ra kai jd n j = lim jc km j = jkj. Qrhsimopoi ntac to Je rhma 3.5 kai to Je rhma 5., kai upoj tontac oti h epif neia e nai suneq c wc proc th metrik Hausdor mporo me na d soume mi de terh ap deixh thc isoperimetrik c anis thtac gi kurt s mata: 5.3 Je rhma. Estw K kurt s ma ston R n pou peri qei to 0 sto eswterik tou, kai D n h mp la gkou jd n j = jkj. T te, n Ap deixh: Estw C S(K). Up rqoun 1 ; : : : ; m S n 1 t toia ste C = S m : : : S 1 (K). S mfwna me to Je rhma 1 (K)) : : m : : : S 1 (K)) Ap thn llh pleur, to Je rhma 5. mac exasfal zei oti up rqei akolouj a fc s g stoiqe wn thc S(K) t toia ste C s! D n. Upoj tontac oti h epif neia e nai suneq c wc proc thn Hausdor metrik sumpera noume oti n ) = lim San m a ak ma efarmog thc Steiner summetrikopo hshc ja d soume mi ap deixh thc anis thtac Brunn - Minkowski (m no gi kurt s mata). H ap deixh e nai basism nh sto Je rhma tou Brunn (to opo o qoume dh apode xei san efarmog thc anis thtac Brunn - Minkowski, bl pe Je rhma.5), kai prohge tai istorik. Me aut n ton tr po g netai kajar to oti h summetrikopo hsh, h anis thta Brunn - Minkowski kai h isoperimetrik anis thta sqhmat zoun nan (enia o) k klo ide n. 5.4 Je rhma tou Brunn. Estw K kurt s ma ston R n kai S n 1. T te, h sun rthsh g(t) = jk \ (? 1 + t)j n 1 e nai ko lh ston for a thc. 33