18.409 A Algorithist s Toolkit October 20, 2009 Lecture 11 Lecturer: Joatha Keler Scribe: Chaithaya Badi 1 Outlie Today we ll itroduce ad discuss Polar of a covex body. Correspodece betwee or fuctios ad origi-syetric bodies (ad see how covex geoetry ca be a powerful tool for fuctioal aalysis). Fritz-Joh s Theore 2 The Polar of a Polytope Give a bouded polytope C R that cotais the origi i its iterior, we ca represet C as C = {x a i x b i, i = 1,..., k}, where b i > 0. Without loss of geerality, by appropriately scalig each costrait, we ca assue b i = 1, i = 1,..., k. Now the polar of C is give by C = cov(a 1,..., a k ). 2.1 Exaples Let C be the square with corers at (1, 1), (1, 1), ( 1, 1), ( 1, 1). The {a i } = {(1, 0), (0, 1), ( 1, 0), (0, 1)}. The polar has corers at (1, 0), (0, 1), ( 1, 0), (0, 1). Note that the polar is a square rotated ad shruk ito a diaod. This polytope is also referred to as the cross polytope. Note that the facets of C becoe the vertices of C ad vice versa. For exaple, the three diesioal cube s polar is the octahedro. Six facets ad eight vertices correspod to eight facets ad six vertices. The size ad shape of a polar teds to be the reverse of that of the origial set. For exaple, a short bulgig rectagle with corers at (100, 3), (100, 3), ( 100, 3), ( 100, 3) would have a tall copressed polar with corers at (±1/100, 0), (0, ±1/3). Also ote that polars of siplices are siplices. 2.2 Properties of a polar Soe of the useful properties of a polar is suarised here. The properties will be illustrated usig pictures. (C ) = C (proof later). If C is origi-syetric, the so is C. If A B the B A. If A is scaled up, the A is scaled dow. If the polar is low-diesioal, that would ea the origial polytope had to be ubouded i soe directios. 11-1
Traslatio has a very drastic effect o the polar. It ca becoe ubouded just by traslatig the polytope. All these properties ca be illustrated usig the pictures below. 3 Polars of Geeral Covex Bodies Ay covex body ca be thought of as the itersectio of a (possibly ifiite) set of half spaces. These are called suportig hyperplaes. Therefore, the polar of a covex body ca be see as the covex hull of a (possibly ifiite) set of poits, coig fro all of the supportig hyperplaes. With this ituitio oe ca guess about the followig : Polar of a sphere is a sphere. Polar of a sphere of radius r is a sphere of radius 1/r. Polar of a ellipse is a ellipse with axes reversed. Defiitio 1 The polar of a covex body C is give by C = {x R x c 1 c C} We observe that this defiitio is equivalet to the previous defiitio. Propositio 2 For a polytope C give by C = {x a i x b i, i = 1,..., k}, the sets C 1 = C 2 where C 1 = {x R x c 1 c C} ad C 2 == cov(a 1,..., a k ). We skip the proof as it is easy to verify that if x C 1 the x C 2 ad vice versa. We will ow prove that (C ) = C. We would be eedig the cocept of a separatig hyperplae for the proof which we itroduce ow. 3.1 Separatig Hyperplaes Give a covex body K R ad a poit p, a separatig hyperplae for K ad p is a hyperplae that has K o oe side of it ad p o the other. More forally, for a vector ν, the hyperplae H = {x ν x = 1} is a separatig hyperplae for K ad p if 1. ν x 1 for all x K, ad 2. ν p 1. Note that if we replace the right had side of both the above coditios by 0 or ay other costat, we get a equivalet forulatio. We call a separatig hyperplae H a strogly separatig hyperplae if the secod iequality is strict. Theore 3 Separatig Hyperplae Theore: If K is a covex body ad p is a poit ot cotaied i K, the there exists a hyperplae that strogly separates the. Proof We ll sketch a outlie of the proof. It ca be ade rigorous. Cosider a poit x K that is the closest to p i l 2 distace. Cosider the plae H that is perpedicular to the lie joiig x to p ad is passig through the idpoit of x ad p. H ust separate K fro p because if there is soe poit of K, say y, that is o the sae side of H as p, the we ca use the covexity of K to coclude that the poit x which is the itersectio of the hyperplae with the lie joiig x ad y is also i K. x is closer to p that x sice px fors the side of a right agled triagle of which xp is the hypoteuse. This cotradicts the assuptio that x is the poit closest to p. 11-2
3.2 Polar of a Polar We ll use the above result to show why the polar of the polar of a covex body is the body itself. Recall that for a covex body K, we had defied its polar K to be {p k p 1 k K}. Theore 4 Let K be a covex body. The K = K. Proof We kow that K = {p k p 1 k K}. Siilarly K = {y p y 1 p k }. Let y be ay poit i K. The, by the defiitio of the polar, for all p K we have that p y 1. The defiitio of the polar of K iplies that y k. Sice this is true for every y K, we coclude that K K. The other directio of the proof is the otrivial oe ad we ll have to use the covexity of the body ad the separatig hyperplae theore. If possible, let y be such that y K ad y K. Sice y K, we have that P y 1 p K. Sice y K, there exists a strogly separatig hyperplae for y ad K. Let it be H = {x v x = 1}. By the defiitio of separatig hyperplae, we have v k 1 k K. Hece, v K. Also, v y > 1 (sice H is a separatig hyperplae), ad we just showed that v K. This cotradicts our assuptio that y K. Hece K K. 4 Nors ad Syetric Covex Bodies We will show how ors ad syetric covex bodies co-exist. This provides us a way to use the results of Covex Geoetry i Fuctioal Aalysis ad vice versa. Recall that a or o R is a ap q : R R such that: 1. q(ax) = aq(x) for a R (hoogeeity) 2. q(x + y) q(x) + q(y) (triagle iequality) 3. q(x) 0 for all x (oegativity) (actually iplied by 1 ad 2) 4. q(x) = 0 if ad oly if x = 0 (positivity) (without this coditios, q is a seior ) Note that give a or, oe ca costruct a covex body. The siplest beig the uit ball B q = {x R q(x) 1}. It is a easy exercise to verify the covexity of B q. Also as we will show ow, give a covex body C, we ca coe up with a or uder which C is the uit ball. Note that C has to be origi syetric. Defiitio 5 The Mikowski fuctioal of a origi syetric covex body C is the ap p C : R defied by p C (x) = if {x λc} λ>0 (We will soeties deote this by x C, because it is a or.) R To prove that this is a or, oe eeds to verify the properties of hoogeeity, triagle iequality etc. These follow fro the covexity of the body. 4.1 Nors, Duals, ad the Polar For ay or q, we ca defie its dual by q (x) = sup v=0 ( v). It is a exercise to see that the uit ball with respect to the dual or of q is the polar of the uit ball with respect to q. This provides us a direct relatio betwee covex geoetry ad fuctioal aalysis. v x q 11-3
The followig pictures allow us to have a geoetric ituitio of the ors ad their duals. 11-4
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5 Baach Mazur Distace Recall fro last tie the defiitio of the Baach Mazur distace betwee two covex bodies: Defiitio 6 Let K ad L be two covex bodies. The Baach Mazur distace d(k, L) is the least positive d R for which theres a liear iage L of L such that L K dl, where dl is the covex body obtaied by ultiplyig every vector i L by the scalar d. Observe that the above defiitio takes ito cosideratio oly the itrisic shape of the body, ad it is idepedet of ay particular choice of coordiate syste. Also observe that the Baach Mazur distace is syetric i it s iput arguets. If L K dl, the by scalig everythig by d, we get that dl dk. Hece K dl dk, which iplies the syetry property. 6 Fritz Joh s Theore Let B 2 deote the -diesioal uit ball. For ay two covex bodies K ad K, let d(k, K ) deote the Baach Mazur distace betwee the. I the rest of this lecture, we ll state ad prove the Fritz Joh s theore. Theore 7 For ay -diesioal, origi-syetric covex body K, d(k, B ). 2 11-6
I other words, the theore states that for every origi-syetric covex body K, there exists soe ellipsoid E such that E K E. We ll prove that the ellipsoid of axial volue that is cotaied i K will satisfy the above cotaiet. Iforally, the theore says that up to a factor of, every covex body looks like a ball. The above boud of is tight for the cube. If we did t require the coditio that K is origi syetric, the the boud would be, which would be tight for a siplex. The theore ca also be rephrased as the followig: There exists a chage of the coordiate basis for which B K 2 B 2. Theore 8 Let K be a origi-syetric covex body. The K cotais a uique ellipsoid of axial volue. Moreover, this largest ellipsoid is B 2 if ad oly if the followig coditios hold: B2 K There are uit vectors u 1, u 2,..., u o the boudary of K ad positive real ubers c 1, c 2,..., c such that 1. i=1 c iu i = 0, ad 2 2 2. for all vectors x, i=1 c i x, u i = x. Sice the u i are uit vectors, they are poits o the covex body K that also belog to the sphere B 2 Also, the first idetity, i.e. i=1 c iu i = 0, is actually redudat, sice for origi syetric bodies it ca be derived fro the secod idetity. This is because for every u i, it s reflectio i the origi is also cotaied i K B 2. The secod idetity says that the cotact poits (of the sphere with K) act soewhat like a orthooral basis. They ca be weighted so that they are copletely isotropic. I other words, the poits are ot cocetrated ear soe proper subspace, but are pretty evely spread out i all directios. Together they ea that the u i ca be weighted so that their ceter of ass is the origi ad their iertia tesor is the idetity. Also, a siple rak arguet shows that there eed to be at least such cotact poits, sice the secod idetity ca oly hold for x i the spa of the u i. 6.1 Proof of Joh s Theore Proof As part of the proof of Joh s Theore, we ll prove the followig thigs: 1. If there exist cotact poits {u i } as required i the stateet of Theore 8, the B is the uique 2 ellipsoid of axial volue that is cotaied i K. 2. If B is the uique ellipsoid of axial volue that is cotaied i K, the there exist poits {u i } 2 such that they satisfy the two idetities i Theore 8. Proof of 1: We are give uit vectors u 1, u 2,..., u o the boudary of K ad positive real ubers c 1, c 2,..., c such that i=1 c iu i = 0, ad for all vectors x, i=1 c i x, u i 2 = x 2. We wish to show that B 2 is the uique ellipsoid of axial volue that is cotaied i K. Observe that it suffices to show that aog all axis-aliged ellipsoids cotaied i K, B 2 is the uique ellipsoid of axial volue. This is because what we are tryig to prove does t etio ay basis ad is oly i ters of dot-products. Hece, sice the stateet will reai true uder rotatios, provig it for axis-aliged ellipsoids is eough. For each u i we have that for all k K, u i k 1. Hece u i K. Let E be ay axis-aliged ellipsoid such that E K. The K E. Hece {u 1, u 2,..., u } E. Sice E is axis-aliged, it is of the for 2 x i {x i=1 α 2 1}. i V ol(e)/v ol(b 2 ) = i=1 α i. Therefore, to show that V ol(e) < V ol(b 2 ), we ust show that i=1 α i < 1 for ay such E which is ot B 2.. 11-7
Observe that E = i=1 α i 2 2 {Y y i 1}. Also, coditio 2 of Theore 8 is equivalet to the follow T ig: i=1 c iu i u i = Id, where Id is the idetity atrix of size. Now, sice u i u i = 1, we have Trace( i=1 c iu i u T i ) = i=1 c i. Sice Trace(Id ) =, this iplies that i=1 c i =. Let e j deote the vector which has a 1 i the i th coordiate ad 0 i the other coordiates. Clearly u i, e j is the j th coordiate of u i. For i i, sice u i E, we get that j=1 α 2 i u i, e j 2 1. Suig it over all i, we get αi 2 u i, e j 2 c i =. i=1 j=1 i=1 2 However, sice by coditio 2 of Theore 8, i=1 u i, e j = e 2 j, we get α 2 By the AM-GM P i=1 i. i=1 α2 i α 2 ) 1/ iequality, we get that ( i=1 i 1, which iplies that i=1 α i 1. Equality oly holds if all the α i are equal. This shows that i=1 α i < 1 for ay such E which is ot B2, copletig the first part of the proof. Proof of 2: Assue that we are give that B 2 is the uique ellipsoid of axial volue that is cotaied i K. We wat to show that for soe, there exist c i ad u i for 1 i (as i the stateet of Theore 2 2 8), such that for all vectors x, i=1 c i x, u i = x. This is equivalet to showig that c i u i u T i = Id. i=1 Also, takig trace of both sides, we get that i=1 c i =. We already observed that for origi-syetric bodies, the coditio that i=1 c iu i = 0, is iplied by the previous coditio. T Let U i = u. Also, observe that we ca view the space of atrices as a vector of 2 i u i real ubers. 2 Hece we ca paraetrize the space of atrices by R T. Hece i=1 c iu i u i = Id eas that Id / is i the covex hull of the U i (recall that the c i are positive ad su to 1). T If possible, let there be o c i, u i such that i=1 c iu i u i = Id. This eas that Id / is ot i the covex hull of the U i. Hece, there ust be a separatig hyperplae H i the space of atrices that separates Id / fro the covex hull of the U i. 2 For two atrices A ad B, let A B deote their dot product i R, i.e. A B = i,j A ij B ij. Thus, the separatig hyperplae is a atrix H such that A cov(u i ), A H 1, ad Id / H < 1. Let t = Trace(H) = H Id. Let H = H t/(id ). The Id / H = Id / (H t/id ) = t/ (Id / t/id ) = 0. Siilarly, sice A cov(u i ), Trace(A) = 1, we get that A H > 0. Hece, H is such that: 1. Trace(H ) = 0, ad 2. H (u i u i T ) > 0 for all i. Now, let E δ = {x R x T (Id +δh T )x 1. For all i, we have u i (Id +δh T )u i = 1+δu i H u i > 1, sice H (u i u T i ) > 0 u T i H u i > 0. Hece u i E δ. Also, sice H (u i u T i ) > 0 for all i, by copactess, there exists ɛ > 0 such that for all atrices w i the ɛ-eighborhood of the set of all u i satisfy H (ww T ) > 0. Hece, by the previous arguet, ay such w is ot cotaied i E δ. Note that whe δ = 0, we get the uit ball B 2. For every δ > 0 we have that for all w i the ɛ eighborhood of the cotact poits of B 2, w E δ. Hece, as we icrease δ cotiuously startig fro 0, the cotiuity of the trasforatio of E δ iplies that for sufficietly sall δ, boudary(k) E δ = φ. Hece ɛ > 0 such that (1 + ɛ )E δ K. Therefore, to coclude the proof, it suffices to show that V ol(e δ V ol(b2 ). Let λ 1, λ 2,..., λ be the eigevalues of Id + δh. Sice V ol(e δ = ( i=1 λ i ) 1, to show that V ol(e δ V ol(b 2 ), we eed to show that i=1 λ i 1. However we kow that i=1 λ i = Trace(Id + δh ) = Trace(Id ) =. By the AM-GM iequality, ( i=1 λ i) 1/ ( i=1 λ i)/ = 1. Hece i=1 λ i 1. This cocludes the proof of part 2. To wrap up the proof of Joh s Theore, assue without loss of geerality that B 2 is the ellipsoid of axial volue cotaied i K. We ca ake this assuptio sice the particular choice of basis is ot 11-8
iportat for the proof. We eed to show that B K 2 B2. Now, for all x K, we have x u i 1 for 2 all i. Hece, x = c u i ) 2 i (x c i =. This shows that x, ad hece K B 2. Thus, we have prove the existece of a ellipse E such that E K E. 11-9
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