An Extremal Property of the Regular Simplex

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2 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 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

3 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.

4 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 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

5 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 , [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, [J] F. Joh, Extremum problems with iequalities as subsidary coditios, Courat Aiversary Volume, Itersciece, New York, 1948, [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 , Michael Schmuckeschläger Istitut für Mathematik Johaes Kepler Uiversität Liz Alteberger Straße 69 A-44 Liz Austria schmucki@caddo.bayou.ui-liz.ac.at