Covex Geometric Aalysis MSRI Publicatios Volume 34, 1998 A Extremal Property of the Regular Simplex MICHAEL SCHMUCKENSCHLÄGER Abstract. If C is a covex body i R such that the ellipsoid of miimal volume cotaiig C the Löwer ellipsoid is the euclidea ball B 2, the the mea width of C is o smaller tha the mea width of a regular simplex iscribed i B 2. 1. Itroductio ad Notatio Suppose that C is a covex body i R such that is a iterior poit of C, the the mea width wc is defied by S 1 wc : = sup x, y if x, y σdx = 2 S 1 sup x, y σdx = 2c sup x, y γ dx R where c is a costat depedig oly o the dimesio, σ the ormalized Haar measure o the sphere S 1 ad γ the -dimesioal stadard gaussia measure. Deotig by C the polar of C with respect to ad by. C the gauge of C, we obtai the well kow formula wc = 2c R x C γ dx =: 2c lc. The euclidea ball B2 is the Löwer ellipsoid of C if ad oly if B 2 is the Joh ellipsoid of C i.e., the ellipsoid of maximal volume cotaied i C. Hece, i order to prove that the regular simplex has miimal mea width, it is eough to prove that for all covex bodies K whose Joh ellipsoid is the euclidea ball, we ecessarily have lk lt, i.e., the l-orm of K is bouded from below by the l-orm of the regular simplex T. The proof of this iequality will follows closely Keith Ball s proof i [B1], where it is show that for ay covex body K there exists a affie image K of K for which the isoperimetric quotiet Vol 1 K/Vol 199 K 1 is o larger
2 MICHAEL SCHMUCKENSCHLÄGER tha the isoperimetric quotiet of a regular simplex. Frack Barthe [B] proved a reversed iequality: amog covex bodies whose Löwer ellipsoid is the euclidea ball the regular simplex has maximal l-orm. 2. The Proof The first igrediet of the proof is a well-kow theorem of F. Joh [J]: Theorem 2.1. Let K be a covex body i R. The the euclidea ball B 2 is the Joh ellipsoid of K if ad oly if there exist uit vectors u j K, 1 j m ad positive umbers c j such that i m c ju j u j = id R ad ii m c ju j =. The secod is a iequality due to Brascamp ad Lieb [BL]. We state this iequality i its ormalized form, as it was itroduced by Ball i [B2]. Theorem 2.2. Let u j, 1 j m, be a sequece of uit vectors i R ad c j positive umbers such that m c ju j u j = id R. The, for all oegative itegrable fuctios f j : R R, m m f j x, u j cj dx R f j cj. Equality holds if, for example, the f j s are idetical gaussias or the u j s form a orthoormal basis. By Joh s theorem there exist uit vectors u j K ad positive umbers c j such that c j u j u j = id R ad c j u j =. Puttig v j := +1 u j, 1 +1 R +1 ad d j = +1 c j it is easily checked that d j v j v j = id R +1 ad d j v j = + 1 Pr +1 1 The first idetity implies d j z, v j 2 = z 2 2 ad d j = + 1. For α R let µ be the measure o R with desity 1 expαt + 1 t 2 /2.
AN EXTREMAL PROPERTY OF THE REGULAR SIMPLEX 21 The by 1 we obtai γ µ [vj ] = m I,] z, v j γ dx e αt +1 γ 1 dt = I,] z, v j d j exp 1 2 R +1 d j z, v j 2 +1 1 exp α d j z, v j dz Puttig fs = 1 e s2 /2 αs I,] s we coclude by the Brascamp Lieb iequality that m γ µ [vj ] = f z, v j d j dz P d j +1 fs ds = fs ds, ad equality holds if the vectors v j form a orthoormal basis i R +1 i.e., if the vectors u j spa a regular simplex. Thus, deotig by u j, 1 j + 1, the cotact poits of a regular simplex T ad the euclidea ball ad by vj the correspodig uit vectors i R +1, the above iequality states that γ µ [vj ] γ µ [v j ] O the other had { } [vj ] = z = x, t R R : t, x t K,. 2 where K := [u j 1] K. Hece we get, by Fubii s theorem, γ µ [vj ] = 1 t γ K e αt +1 t 2 /2 dt. Now, sice K K, this implies by 2, 1 t γ K e λt t2 /2 dt 1 ad therefore 1 γ. K e λt t2 /2 dt 1 λ2 /2 Multiplyig both sides by e ad Fubii s theorem, γ. K dt γ t T e λt t2 /2 dt, γ. T e λt t2 /2 dt. itegratig over λ R we obtai, by γ. T > t dt
22 MICHAEL SCHMUCKENSCHLÄGER from which we readily deduce that lk lt. More geerally we get, for each o egative fuctio ϕ, γ. K > t R ϕt xe x2 /2 dx dt γ. T > t R ϕt xe x2 /2 dx dt. Remark. If we restrict the problem to covex ad symmetric bodies, the we get a iequality for the distributio fuctio see [SS]: For all covex symmetric bodies B i R whose Joh ellipsoid is the euclidea ball we have, for all t >, γ. B > t γ. > t. Refereces [B] F. Barthe, Iégalités de Brascamp Lieb et covexité, to appear i Comptes Redus Acad. Sci. Paris. [BL] H. J. Brascamp ad E. H. Lieb,Best costats i Youg s iequality, its coverse, ad its geeralizatio to more tha tree fuctios, Advaces i Math. 2 1976, 151 173. [B1] K. M. Ball, Volume ratios ad a reverse isoperimetric iequality. [B2] K. M. Ball, Volumes of sectios of cubes ad related problems, GAFA Semiar, Lecture Notes i Mathematics 1376, Spriger, 1989, 251 26. [J] F. Joh, Extremum problems with iequalities as subsidary coditios, Courat Aiversary Volume, Itersciece, New York, 1948, 187 24. [SS] G. Schechtma ad M. Schmuckeschläger, A cocetratio iequality for harmoic measures o the sphere, GAFA Semiar ed. by J. Lidestrauss ad V. Milma, Operator Theory Advaces ad Applicatios 77 1995, 255 274. Michael Schmuckeschläger Istitut für Mathematik Johaes Kepler Uiversität Liz Alteberger Straße 69 A-44 Liz Austria schmucki@caddo.bayou.ui-liz.ac.at