THE JOHN ELLIPSOID THEOREM. The following is a lecture given in the functional analysis seminar at the University of South Carolina.

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1 THE JOHN ELLIPSOID THEOREM RALPH HOWARD DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, SC 29208, USA The following is lecture given in the functionl nlysis seminr t the University of South Crolin Contents Introduction 2 Proof of the theorem 2 3 The cse of centrlly symmetric convex bodies 6 4 Proof of the uniquness of the John ellipsoid 7 4 Exmples of where the inequlities re shrp 0 References 0 Introduction An ffine mp Φ: R n R n is mp of the form Φx) :=Ax + b where A: R n R n is liner nd b R n is constnt vector This is nonsingulr iff det A 0 Anellipsoid in R n the imge of the closed unit bll B n of R n under nonsingulr ffine mp Our gol here is to prove the following fmous result of Fritz John Theorem John [3]) Let K R n be convex body tht is compct convex set with nonempty interior) Then there is n ellipsoid E clled the John ellipsoid which will turn out to be the ellipsoid of mximl volume contined in K) so tht if c is the center of E then the inclusions E K c + ne c) hold Here c + ne c) is the set of points {c + nx c) :x E} This is the diltion of E by fctor of n with center c) Dte: November 997

2 2 RALPH HOWARD Remr This result is shrp in the sense tht the diltion fctor n cn not be replced by smller fctor As n exmple let x 0,,x n be n ffinely independent set of points in R n nd let K be the convex hull of {x 0,,x n } Tht is K is n n dimensionl simplex in R n Then if E is the ellipsoid of mximl volume in K then E K c + ne c), but there is no ellipsoid E K so tht E K c + me c ) for ny rel number m<n For the outline the proof of these clims see Section 4 For nther exmple see Remr 23 The proof here is one I cme up with bsed loosely on ides in the expository rticle [2] of Berger where he gives similr proof in the cse K is symmetric bout the origin For different proof, which in mny wys is preferble to the bre hnded pproch here, see the rticle of Keith Bll [] John originl proof [3] nd I thn Steve Dilworth for giving me copy of this pper) is quite different nd deduces the result from more generl result on mxim nd minim of functions subject to inequlity constrints very much in spirit of wht is now clled geometric progrmming 2 Proof of the theorem For the rest of this section K will be convex body in R n The bsic ide of the proof is to choose E R n to be n ellipsoid of mximl volume Then by n ffine chnge of vribles we cn ssume tht E is the unit bll B n The proof is completed by showing tht if K contins point p t distnce greter thn n from the origin of R n then the convex hull of B n {p}, nd thus lso K, contins n ellipsoid of volume greter thn B n which would contrdict tht B n hs mximl volume See Figure B n E If p is lrge enough then conv{b n,p} will contin n ellipsoid E with VolE) > VolB n ) p Figure We strt by showing tht K contins n ellipsoid of mximl volume If Φx) =Ax + b is n ffine mx then the volume of the ellipsoid E =Φ[B n ] is VolE) = deta) VolB n ) This is stndrd piece of ffine geometry, but it cn lso be deduced from the chnge of vrible formul for integrls Lemm 2 The convex body K contins n ellipsoid of mximl volume Proof Let N := n + n 2 Then the set of ordered pirs A, b) where A is n n n mtrix nd b R n is just R N Let E := {A, b) R N : AB n + b

3 THE JOHN ELLIPSOID THEOREM 3 K} Then E is closed bounded nd thus compct subset of R N nd A, b) deta) is continuous function on E Thus there is n A 0,b 0 ) tht mximizes this function on E Then E := A 0 B n + b 0 is the desired ellipsoid If E = A 0 B n +b 0 is n ellipsoid of mximl volume in K then by replcing K by A 0 K b 0) we cn ssume tht B n = A 0 E b 0) is n ellipsoid of mximl volume in K Then to prove John s Theorem it is enough to show tht if p K then p nwhere is the stndrd Eucliden norm) The geometric ide of the proof is shown in Figure If p is lrge then the convex hull conv{b n,p} of B n nd p will contin n ellipsoid E with VolE) > VolB n ) But K is convex so E conv{b n,p} Kwhich contrdicts tht B n is n ellipsoid of mximl volume in K Wht tes some wor is showing the criticl distnce where B n stops hving mximl volume in conv{b n,p} is p >n x= B 2 Φ λ t[b 2 ] Figure 2 We will construct the ellipsoid E s n ffine imge of the bll B n This will first be done in two dimensions nd then extended to higher dimensions by symmetry As it is truth universlly cnowledged tht ny problem in nlysis needs differentil eqution to be ten seriously, we will construct our ffine mps s flows of solutions to differentil equtions Let λ>0 then in the plne consider the system of equtions ẋ =+x ẏ= λy The solution with initil condition x0) = x 0 nd y0) = y 0 is xt) = +e t + x 0 ) yt) =e λt y 0 Therefore if Φ λ t is the one prmeter group of diffeomorphisms generted by this system of equtions or wht is the sme thing the one prmeter group generted by the vector field + x) x λ y )is Φ λ tx, y) = +e t + x),e λt y)= +e t,0)+e t x, e λt y)

4 4 RALPH HOWARD so tht for ech fixed t the mp x, y) Φ λ t x, y) is n ffine mp The lines x = nd y = 0 re fixed set-wise) by Φ λ t The effect of Φ λ t on the two dimensionl bll B 2 for smll positive vlue of t is shown in Figure 2 For > let C := conv{b 2,, 0)} be the convex hull of the two dimension bll B 2 nd the point, 0) See Figure 3 B 2, 0) C is convex hull of B 2 nd, 0) Figure 3 The following is the hrd step in the proof of John s theorem Lemm 22 If λ> then Φλ t [B 2 ] C for smll positive vlues of t Proof We first note tht the tngent lines to B 2 through, 0) which re prt of the boundry of C ) re y = ± x ) To see this consider the 2 line y = x ) Direct clcultion shows tht both the points, 0) 2 ) ) nd, 2 re on this line nd the point, 2 is on the unit circle see Figure 4) The slope of the rdius to B 2 ) ending t, 2 is 2 ) The slope of y = x ) is 2 which is the negtive 2 reciprocl of the slope of the redius to B 2 ) ending t, 2 Therefore y = x ) is prependiculr to the rdius where it meets 2 B2 nd this is tngent ), 2 y = x ) 2, 0) Figure 4 Therefore to prove the lemm it is enough to show tht for λ> nd smll t tht Φ λ ) t, 2 lies below the line y = x ) For s 2

5 B 2 is tngent to this line t Φ λ t See Figure 5) ), 2 THE JOHN ELLIPSOID THEOREM 5 Φ λ t, 2, 2 ) it will lso be cried into C by ) Figure 5 Either directly from the definition of Φ λ t or from the system of differentil equtions defining it we hve d dt Φλ t, 2 ) =ẋ0), ẏ0)) = +, λ 2 ) t=0 Therefore the slope of the tngent to the flow line t Φ λ t, ) 2 t, ) 2 is ẏ0) ẋ0) = λ 2 + Therfore Φ λ t, ) 2 will be below the line y = x ) provided 2 λ 2 + < 2 A little lgebr shows this is equivlent to λ> proof of the lemm This completes the Proof of Thoerem Assume tht B n K is n ellipsoid of mximl volume in K Then we need to show tht then is not point p of K with p > n Assume, towrd contrdiction, tht there is such point Choose orthogonl coordintes x, y,,y n )onr n so tht p is on the positive x xis Then if = p then in these coordintes p =, 0,,0) The higher dimensionl version of C used bove it C n := conv{b n,p} = conv{b n,, 0,,0)} Define the higher dimensionl nlogue Φ λ t, tht is Ψ λ t x, y,,y n )= +e t x+),e λt y,,e λt y n ) As before this is n ffine mp From Lemm 22 nd symmetry we hve tht Ψ λ t [B n ] C n K for smll t provided λ> The volume of the ellipsoid Ψ λ t [B n ]is VolΨ λ t [B n ]) = e n )λ)t VolB n )

6 6 RALPH HOWARD Now if >nthen it is possible to choose λ with λ> nd n )λ >0 This implies for smll positive t tht Ψ λ t [B n ] C n K nd VolΨ λ t [B n ]) > VolB n ) This contrdiction completes the proof of John s theorem Remr 23 The proof of the theorem gives nther exmple showing tht the diltion fctor n is shrp in John s theorem If = n then B n will be the ellipsoid of mximl volume in C nd C αb n only for α n 3 The cse of centrlly symmetric convex bodies When K is symmetric bout the origin, tht is K = K, then the diltion fctor in John s Theorem cn be improved Theorem 2 Let K be convex body in R n which is symmetric bout the origin Then there is n ellipsoid E, lso symmetric bout the origin, so tht E K ne Remr 3 The fctor n is this result is shrp As n exmple consider K := {x,,x n ) R n : x i }bethe unit bll of l n Then the unit bll B n is the ellipsoid of mximl volume in K nd the point P :=,,,) K hs p = n which shows tht n is best possible Remr 32 Let X, X ) nd Y, Y ) be two finite n-dimensionl vector spces Then define the Bnch-Mzur distnce between the two spces s dist BM X, Y ) := inf T T T : X Y where T rnges over ll liner isomorphisms of T : X Y nd is the opertor norm Then resttement of Theorem 2 is tht for nd n dimensionl X tht dist BMX, l 2 n) n nd Remr 3 implies tht dist BM l n,l 2 n) = distl n,l 2 n)= nso tht this is the shrp bound Proof The proof is very much lie the generl cse A vrint on Lemm 2 shows there s symmetric ellipsoid E K tht hs mximl volume By n ffine chnge of coordintes we cn ssume tht the bll B n is the ellipsoid of mximl volume in K Agin we strt by looing t two dimensionl problem This time let S := convb 2 {, 0),, 0)} See Figure 6) As we wish to preserve the symmetry of figures this time we loo t the flow of the equtions ẋ = x ẏ = λy where λ>0 This hs s flow Φ λ t x, y) =e t x, e λt y) which is liner Agin we wnt to now when Φ λ t [B 2 ] S for smll t As ) in the proof of Lemm 22 looing t wht hppens to the point, 2

7 , THE JOHN ELLIPSOID THEOREM 7 ) ) 2, 2 B 2 y = 2 x ), 0), 0), 2 ), 2 ) Figure 6 is the ey We hve d dt Φλ t, 2 ) =ẋ0), ẏ0)) =, λ 2 t=0 Thus the condition for Φ λ ) t, 2 to be below the line y = for smll t is tht ẏ0) ẏ0) = λ 2 < 2 ) x ) 2 Some lgebr then shows tht Φ λ t [B 2 ] S for smll t if λ? 2 To complete the proof if K is symmetric convex body so tht B n is the ellipsoid of mximum volume contined in K then we wish to show tht ny p K stisfies p n We cn choose coordintes x, y,,y n so tht p is on the positive x-xis Then p =, 0,,0) By symmetry p =, 0,,0) is lso in K Let S n = convb n {p, p}) nd define one prmeter group of liner mps cting on R n by Ψ λ t x, y,,y n )=e t x, e λt y,,e λt y n ) Then by symmetry Ψ λ t [B n ] S K for smll t if λ> The volume 2 of Ψ λ t [B n ]is VolΨ λ t [B n ]) = e n )λ)t VolB n ) Now if = p > n then it is possible to choose λ so tht λ> 2 so tht Ψ λ t [B n ] S K for smll t) nd n )λ >0 so tht VolΨ λ t [B n ]) > VolB n ) This is contrdiction nd completes the proof of Theorem 2 4 Proof of the uniquness of the John ellipsoid Theorem 3 If K R n is compct body then the ellipsoid of mximl volume is unique Lemm 4 Let E be n ellipsoind in R n, so tht E = AB n + b with A liner nd b R n Then in this representtion we cn te A to be positive definite

8 8 RALPH HOWARD Proof It is stndrd result from liner lgebr the polr decomposition ) tht ny nonsingulr mtrix cn be expresses s product A = PU where P is positive definite nd U is orthogonl For ny orthogonl mtrix we hve UB n = B n Thus AB n = PUB n = PB n Thus we cn replce A by P in the representtion E = AB n + b nd ssume tht A is positive definite Proposition 42 Let A nd B be n n positive definite mtrices Then deta + B) n deta) n + detb) n with equlity iff there is positive constnt c so tht B = ca In prticulr if equlity holds nd deta) = detb) then A = B Proof As A is positive definite it hs positive definite squre root P Tht is A = P 2 Then deta + B) n = detp 2 + B) n = detp ) 2 n deti + P BP ) n nd deta) n + detb) n = detp ) 2 n + detp AP ) n ) Thus letting C = P BP it is enough to show deti +C) n +detc) n with equlity iff C = ci If C = ci then P BP = ci so tht B = cp 2 = ca) Now let λ,,λ n be the eigenvlues of C Then deti + C) n + detc) b is equivlent to 4) n = ) n + λ ) +λ λ n ) n with equlity iff λ = c for ll nd some positive c Let σ = σ λ,,λ n ) be the elementry symmetric function of λ,,λ n Tht is Then σ λ,,λ n ):= i <i 2 < <i n λ i λ i2 λ i n + λ )=+σ +σ 2 + +σ n =

9 THE JOHN ELLIPSOID THEOREM 9 Note σ n = λ λ n ) By the inequlity between the rithmetic nd geometric mens we hve ) n λ i λ i2 λ σ λ,,λ n )= ) i n i <i 2 < <i n ) n λ i λ i2 λ i ) n ) i <i 2 < <i n ) n ) n = λ λ 2 λ n ) n ) ) n = λ λ 2 λ n ) n nd equlity will hold iff λ i λ i2 λ i =λ j λ j2 λ j for ll subsets {i,,i } nd {i,,i } of {,,n} of size This is the cse iff there is c so tht λ i = c for ll i Using this inequlity we hve n ) +λ λ n ) n n ) n = λ λ n ) n +σ +σ 2 + +σ n n = + λ ) = with equlity iff λ = c for some c>0 But this inequlity is equivlent to the inequlity 4) This completes the proof Proof of Theorem 3 As bove let P n be the set of n n positive definite mtrices nd let K := P n R n Let E := {A, b) K:AB n + b K} By Lemm 4 every ellipsoid contined in K is of the form AB n + b for some A, b) K Also s K is convex the set E is lso convex Let E = A B n +b nd E 2 = A 2 B n + b 2 be two ellipsoids of mximl volume contined in K The volume of the ellipsoid AB n + b is deta) VolB n ), so s E nd E 2 re both mximl we hve deta ) = deta 2 ) Let A 3 = 2 A + A 2 ) nd b 3 = 2 b + b 2 ) Then s E is convex the ellipsoid E 3 = A 3 B n + b 3 K But from Proposition 42 deta 3 ) n = 2 deta + A 2 ) n 2 deta ) n + deta2 ) n ) = deta ) n But s E hs mximl volume this implies deta 3 ) = deta ) = deta 2 ) so tht equlity holds in the inequlity By Proposition 42 this implies A = A 2 Tht is E 2 is trnslte of E If only remins to show b = b 2 Ifb b 2 then K will contin the convex hull of E E 2 Let b 3 = 2 b + b 2 ) Then conve E 2 ) will contin n ellipsoid E 3 centered nd b 3 tht hs volume VolE 3 ) > VolE ) s it will =0

10 0 RALPH HOWARD E E 2 E 3 b b 3 b 2 E 3 contins trnslte of E nd thus hs greter volume Trnslte of E centered t b 3 Figure 7 contin trnslte of E centered t b 3 see Figure 7 where we hve done n ffine trnsformtion so tht we cn ssume tht E nd E 2 re blls) But this contrdicts tht E hd mximl volume nd so b = b 2 This completes the proof 4 Exmples of where the inequlities re shrp Above we we mentioned some exmples there the inequlities re shrp However to show tht this is the cse by direct clcultion is not esy so we show here tht the uniqueness result Theorem 3 cn be used to find the ellipsoid of mximum volume in cse where symmetry is present Consider the cse of the stndrd regulr simplex σ n The esiest reliztion of this is to view the ffine spce R n s the hyperplne in R n+ defined by x +x 2 + +x n+ = Then let e,,e n+ be the stndrd bsis of R n+ Then σ n is the convex hull of e,,e n+ Tht is σ is the subset of R n+ defined by x + +x n+ = nd x i 0 Let S n+ be the permuttion group on n + objects Then S n+ cts on σ n R n by permuting its vertices e,,e n+ This ction is by ffine mps nd in fct by isometries This implies tht this ction mps ellipsoids to ellipsoids nd tht it preserves their volume As the ellipsoid of mximl volume in σ n is unique this implies tht it is left fixed by ll elements of S n+ However it is not hrd to chec tht the only ellipsoids in R n invrint under ll elements re the blls centered t n+,, n+ ) Using this nd tht ll simplexes re ffinely equivlent it is not hrd to verify the clims in Remr The clims of Remr 3 cn be checed in similr mnner References Keith Bll, Ellipsoids of mximl volume in convex bodies, Geom Dedict 4 992), no 2, Mrcel Berger, Convexity, Amer Mth Monthly ), no 8, Fritz John, Extremum problems with inequlities s subsidiry conditions, Studies nd Essys Presented to R Cournt on his 60th Birthdy, Jnury 8, 948, Interscience Publishers, Inc, New Yor, N Y, 948, pp