The Interfac e betwee n Convex Geometr y an d

Harmonie Analysi s

This page intentionally left blank

Conference Boar d o f the Mathematica l Science s

CBMS Regional Conference Series in Mathematics

Number 10 8

The Interfac e betwee n Convex Geometr y an d

Harmonie Analysi s

Alexander Koldobsk y Vladyslav Yaski n

Published fo r th e Conference Boar d o f th e Mathematica l Science s

by th e American Mathematica l Societ y S**dbf%

Providence, Rhod e Islan d ±-£am#* wlth suppor t fro m th e

%ef3* Nationa l Scienc e Foundatio n °»otV

http://dx.doi.org/10.1090/cbms/108

NSF-CBMS Regiona l Conferenc e o n

The Interfac e betwee n Conve x Geometr y an d Harmoni e Analysi s

heia a t Kansa s Stat e University , Manha t tan , KS ,

July 29-Augus t 3 , 2006 .

Partially supporte d b y th e Nationa l Scienc e Foundation .

The author s acknowledg e suppor t fro m th e Conferenc e Boar d o f Mathematical Science s an d NS F Gran t DMS-0532656 .

2000 Mathematics Subject Classification. Primar y 52A20 , 42A38 , 44A05 .

For additiona l informatio n an d Update s o n thi s book , visi t w w w . a m s . o r g / b o o k p a g e s / c b m s - 1 0 8

Library o f Congres s Cataloging-in-Publicat io n D a t a

Koldobsky, Alexander , 1955-The interfac e betwee n conve x geometr y an d harmoni c analysi s / Alexande r Koldobsky ,

Vladyslav Yaskin . p. cm . — (Regiona l Conferenc e serie s i n mathematics, ISS N 0160-764 2 : no . 108 )

Includes bibliographica l reference s an d index . ISBN-13: 978-0-8218-4456- 4 (alk . paper) ISBN-10: 0-8218-4456- 3 (alk . paper) 1. Conve x geometry—Congresses . 2 . Harmonic analysis—Congresses . I . Yaskin , Vladyslav ,

1974- II . Title .

QA639.5.K64 200 8 516'.08—dc22 200706057 2

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them, ar e permitted t o make fai r us e of the material, suc h a s to copy a chapte r fo r use in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t o f the source i s given .

Republication, systemati c copying , or multiple reproduetio n o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed t o the Acquisitions Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Requests ca n als o b e mad e b y e-mail t o reprint-permission@ams.org .

© 200 8 b y the American Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s

except thos e grante d t o the United State s Government . Printed i n the United State s o f America .

@ Th e paper use d i n this boo k i s aeid-free an d falls withi n th e guideline s established t o ensure permanenc e an d durability .

Visit th e AMS hom e pag e a t http://www.ams.org /

10 9 8 7 6 5 4 3 2 1 1 3 12 11 10 09 0 8

To Olga and Marin a

This page intentionally left blank

Contents

Preface i x

Chapter 1 . Hyperplan e section s o f £p-bal\s 1 1.1. Lectur e 1 1

Chapter 2 . Volum e an d th e Fourie r transfor m 9 2.1. Lectur e 2 9

Chapter 3 . Intersectio n bodie s 2 1 3.1. Lectur e 3 2 1 3.2. Intersectio n bodie s an d ellipsoid s 2 6 3.3. Non-intersectio n bodie s al l o f whos e centra l section s ar e

intersection bodie s 3 0 3.4. Generalize d intersectio n bodie s 3 3

Chapter 4 . Th e Busemann-Pett y proble m 3 9 4.1. Lectur e 4 3 9 4.2. A modification o f the Busemann-Pett y proble m 4 3 4.3. Th e Busemann-Pett y proble m i n hyperboli c an d spherica l

Spaces 4 5 4.4. Th e lowe r dimensiona l Busemann-Pett y proble m 5 5

Chapter 5 . Projection s an d th e Fourie r transfor m 5 9 5.1. Lectur e 5 5 9

Chapter 6 . Intersectio n bodie s an d L p-spaces 6 7 6.1. Lectur e 6 6 7

Chapter 7 . O n th e roa d betwee n pola r projectio n bodie s an d intersection bodie s 7 5

7.1. Lectur e 7 7 5 7.2. Th e geometr y o f L 0 7 7 7.3. A Banach subspac e o f L 1/2 tha t doe s no t embe d i n L\ 8 0 7.4. Commo n subspace s o f L p Space s 8 4

Chapter 8 . Ope n problem s 8 7

Bibliography 10 1

Index 10 7

This page intentionally left blank

Preface

This tex t arise s fro m te n lecture s give n b y th e first-named autho r a t the NSF/CBM S Conferenc e "Th e Interfac e betwee n Conve x Geometr y an d Harmonie Analysis " hel d o n Jul y 29-Augus t 3 , 200 6 a t Kansa s Stat e Uni -versity i n Manhattan , KS . Th e mai n topi c o f thes e lecture s i s th e Fourie r analytic approac h t o th e geometr y o f convex bodies develope d ove r th e las t few years . Th e ide a o f thi s approac h i s t o expres s certai n geometri c prop -erties o f conve x bodie s i n term s o f th e Fourie r transfor m an d the n appl y methods o f harmonic analysi s to solve geometric problems . Th e Fourie r ap -proach has led to severa l important results , includin g a n analyti c Solutio n of the Busemann-Petty proble m on sections of convex bodies, characterization s of intersectio n bodie s an d thei r connection s wit h th e theor y o f L p-spaces, extremal section s an d projeetion s o f certain classe s o f bodies , an d a unifie d approach t o section s an d projeetion s o f convex bodies .

The main feature s o f the Fourie r approac h t o convexity were reflected i n the book "Fourie r Analysi s in Convex Geometry" writte n b y the first-name d author an d publishe d i n 2005 in the Mathematica l Survey s and Monograph s series o f th e America n Mathematica l Society . Tha t boo k include s rigorou s proofs o f al l mai n result s tha t appeare d befor e th e yea r 2005 , a s wel l a s short description s o f th e mai n tool s fro m convexity , Fourie r analysis , in -tegral geometr y an d probabilit y use d i n th e proof s o f thes e results . Th e main purpos e o f th e curren t boo k i s different ; i t expose s i n a shor t for m the mai n idea s o f th e Fourie r approac h t o geometr y s o tha t intereste d re -searchers an d student s ca n quickl y lear n th e subjee t an d star t workin g o n related problems . Beyon d that , w e include her e severa l interestin g ne w re -sults tha t hav e appeare d afte r th e boo k [K9] , i n particula r th e Solutio n o f the Busemann-Pett y proble m i n non-Euclidea n Spaces , non-equivalenc e o f several generalization s o f intersectio n bodies , ne w method s o f construetin g non-intersection bodies , and a continuous path betwee n intersection an d po-lar protectio n bodie s leadin g t o som e insight s abou t th e mysteriou s dualit y between sections and projeetions o f convex bodies. Th e last chapte r include s several ope n problem s an d discussion s o f related results .

The struetur e o f th e boo k i s a s follows . Ever y chapte r Start s wit h a section includin g the actua l lectur e give n a t th e Conference . W e recommen d that beginner s star t b y readin g th e first sectio n o f eac h chapte r only . Thi s way th e reade r get s t o kno w th e mai n definitions , coneepts , an d results . The detail s o f proof s ar e sometime s no t give n - w e prefe r t o expos e th e

ix

X PREFACE

main underlyin g idea s - bu t i n ever y suc h plac e w e giv e reference s t o th e book [K9 ] or to one of the secondary sections of this book. Afte r readin g th e first section s o f al l chapters , th e reade r wil l b e prepare d fo r othe r section s of this boo k an d fo r relate d paper s no t include d i n the text .

Finally, we would like to thank the Organizers of the Conference a t Kansa s State - David Auckl y and Dmitr i Ryabogin - for the wonderful Job of putting together a meeting that include d experienced researchers , young researchers, graduate students , an d undergraduat e students . Th e additiona l session s that the y organize d fo r th e student s wer e ver y helpful . Th e first-named author acknowledge s th e suppor t o f th e U.S . Nationa l Scienc e Foundatio n through th e grant s DMS-045569 6 an d DMS-0652571 . Th e second-name d author acknowledge s the support o f the European Network PHD, FP6 Mari e Curie Actions, RTN, Contract MCRN-511953 , and the U.S. National Scienc e Foundation, gran t DMS-0455696 .

Alexander Koldobsk y an d Vladysla v Yaski n

This page intentionally left blank

Bibliography

[A] Yu . A. Aminov, The geometry of submanifolds, Gordo n an d Breach Scienc e Publishers , Amsterdam, 2001 .

[Bai] K . Ball , Cube slicing in W 1, Proc . Amer . Math . Soc . 9 7 (1986) , 465-473 . [Ba2] K . Ball , Shadows of convex bodies, Trans. Amer . Math . Soc . 32 7 (1991 ) , 891-901 . [Ba3] K . Ball , Some remarks on the geometry of convex sets, in : Geometri e Aspect s o f

Functional Analysis , ed . b y J . Lindenstraus s an d V . D . Milman , Lectur e Note s i n Math. 1317 , Springer , Heidelberg , 1988 , 224-231 .

[Ba4] K . Ball , Normed Spaces with a weak-Gordon-Lewis property, Functiona l analysi s (Austin, TX , 1987/1989) , 36-47 , Lectur e Note s i n Math . 1470 , Springer , Berlin , 1991.

[Bar] F . Barthe , Extremal properties of central half-Spaces for produet measures, J . Funct . Anal. 18 2 (2001) , 81-107 .

[Bol] E . D. Bolker, A dass of convex bodies, Trans. Amer. Math . Soc . 145 (1969) , 323-345. [Bol] J . Bourgain , On the distribution of polynomials on high dimensional convex sets,

Geometrie aspect s o f functiona l analysi s (1989-90) , 127-137 , Lectur e Note s i n Math . 1469, Springer , Berlin .

[Bo2] J . Bourgain , On the Busemann-Petty problem for perturbations of the ball, Geom . Func. Anal . 1 (1991) , 1-13 .

[BDK] J . Bretagnolle , D . Dacunha-Castell e an d J . L . Krivine , Lois stables et espaces L p, Ann. Inst . H.Poincar e Probab . Statist . 2 (1966 ) 231-259 .

[BK] F . Barthe , A . Koldobsky , Extremal slabs in the cube and the Laplace transform, Adv. Math . 17 4 (2003) , 89-114 .

[BKM] J . Bourgain , B . Klartag , V . Milman , Symmetrization and isotropic constants of convex bodies, Geometri e aspect s o f functiona l analysis , 101-115 , Lectur e Note s i n Math. 1850 , Springer , Berlin , 2004 .

[BL] J . Barker , D . Larman , Determination of convex bodies by certain sets of sectional volumes, Discret e Math . 24 1 (2001) , no . 1-3 , 79-96 .

[BN] F . Barth e an d A . Naor , Hyperplane projeetions of the unit ball ofl™, Discrete Com -put. Geom . 2 7 (2002) , 215-226 .

[BP] H . Busemann, C . M. Petty, Problems on convex bodies, Math. Scand . 4 (1956) , 88-94. [Bu] H . Busemann , A theorem on convex bodies of the Brunn-Minkowski type, Proc . Nat .

Acad. Sei . U.S.A . 3 5 (1949) , 27-31 . [BZ] J . Bourgain , Gaoyon g Zhang , On a generalization of the Busemann-Petty problem,

Convex geometri c analysi s (Berkeley , CA , 1996) , 65-76 , Math . Sei . Res . Inst . Publ . 34, Cambridg e Univ.Press , Cambridge , 1999 .

[C] A . M . Caetano , Weyl numbers in sequence Spaces and sections of unit balls, J. Funct . Anal. 10 6 (1992) , 1-17 .

[DFN] B . A . Dubrovin , A . T . Fomenko , S . P . Novikov , Modern geometry - methods and applications. Part I. The geometry of surfaces, transformation groups, and fields. Sec-ond edition, Springer-Verlag , Ne w York , 1992 .

[Fa] K . J . Falconer , X-ray problems for point sources, Proc . London Math . Soc . 46 (1983) , 241-262.

102 BIBLIOGRAPHY

[Fei] W . Feller , An introduction to probabüity theory and its applications, Wile y & Sons , New York , 1971 .

[Fer] T . S . Ferguson , A representation of the Symmetrie bivariate Cauchy distributions, Ann. Math . Stat . 33 , (1962) , 1256-1266 .

[FGW] H . Fallert , P . Goode y an d W . Weil , Spherical projeetions and centrally Symmetrie sets, Adv . Math . 12 9 (1997) , 301-322 .

[Gal] R . J . Gardner , Symmetrals and X-rays of planar convex bodies, Arch . Math . 4 1 (1983), 183-189 .

[Ga2] R . J . Gardner , Intersection bodies and the Busemann-Petty problem, Trans . Amer . Math. Soc . 34 2 (1994) , 435-445 .

[Ga3] R . J . Gardner , A positive answer to the Busemann-Petty problem in three dimen-sions, Annai s o f Math . 14 0 (1994) , 435-447 .

[Ga4] R . J . Gardner , Geometrie tomography, Cambridg e Universit y Press , Cambridge , 1995.

[Gi] A . Giannopoulos , A note on a problem of H. Busemann and C. M. Petty concerning sections of Symmetrie convex bodies, Mathematika, 3 7 (1990) , 239-244 .

[GKS] R . J . Gardner , A . Koldobsky , T . Schlumprecht , An analytic Solution to the Busemann-Petty problem on sections of convex bodies, Annai s o f Math . 14 9 (1999) , 691-703.

[Gr] H . Groemer , Geometrie applications of Fourier series and spherical harmonics, Cam -bridge Universit y Press , Ne w York , 1996 .

[GrZ] E . Grinberg , G . Zhang , Convolutions, transforms, and convex bodies, Proc. Londo n Math. Soc . 78(3 ) (1999 ) 77-115 .

[GS] I . M . Gelfan d an d G . E . Shilov , Generalized funetions, voll Properties and Opera-tions, Academi c Press , Ne w Yor k an d London , 1964 .

[GW] P . Goodey an d W . Weil , Intersection bodies and ellipsoids, Mathematik a 4 2 (1995) , 295-304.

[GZ] P . Goode y an d G . Zhang , Inequalities between protection funetions of convex bodies, Amer. J . Math . 12 0 (1998) , 345-367 .

[Ha] H . Hadwiger , Gitterperiodische Punktmengen und Isoperimetrie, Monatsh . Math . 7 6 (1972), 410-418 .

[He] D . Hensley, Slicing the cube in R n and probabüity (bounds for the measure of a central cube slice in R n by probabüity methods), Proc . Amer . Math . Soc . 7 3 (1979) , 95-100 .

[Her] C . Herz , A dass of negative definite funetions, Proc . Amer . Math . Soc . 1 4 (1963) , 670-676.

[J] M . Junge , Hyperplane conjeeture for quotient Spaces of L p, Foru m Math . 6 (1994) , no. 5 , 617-635 .

[Kl] A . Koldobsky , The Schoenberg problem on positive-definite funetions, Algebr a i Analiz 3 (1991) , 78-85 ; translation i n St . Petersbur g Math . J . 3 (1992) , 563-570 .

[K2] A . Koldobsky , Common subspaces of Lp-spaces, Proc . Amer . Math . Soc . 12 2 (1994) , no. 1 , 207-212 .

[K3] A . Koldobsky , A Banach subspace of Li/ 2 which does not embed in L± (isometric version), Proc . Amer . Math . Soc . 12 4 (1996) , no . 1 , 155-160 .

[K4] A . Koldobsky , An application of the Fourier transform to sections of star bodies, Israel J . Math . 10 6 (1998) , 157-164 .

[K5] A . Koldobsky , Intersection bodies, positive definite distributions and the Busemann-Petty problem, Amer . J . Math . 12 0 (1998) , 827-840 .

[K6] A . Koldobsky , Intersection bodies in R 4, Adv . Math . 13 6 (1998) , 1-14 . [K7] A . Koldobsky , Positive definite distributions and subspaces of L- p with applications

to stable processes, Canad . Math . Bull. , 4 2 (1999) , no.3 , 344-353. [K8] A . Koldobsky , A functional analytic approach to intersection bodies, Geom . Funct .

Anal. 1 0 (2000) , 1507-1526 .

BIBLIOGRAPHY 103

[K9] A . Koldobsky , Fourier Analysis in Convex Geometry, Mathematica l Survey s an d Monographs, America n Mathematica l Society , Providenc e RI , 2005 .

[KK1] N . Kaiton , A . Koldobsky , Banach Spaces embedding isometrically into L p when 0 < p < 1 , Proc . Amer . Math . Soc . 13 2 (2004) , no . 1 , 67-7 6

[KK2] N . Kaiton , A . Koldobsky , Intersection bodies and L p-spaces, Adv . Math . 19 6 (2005), 257-275 .

[KKYY] N . J . Kaiton , A . Koldobsky , V . Yaski n an d M . Yaskina , The geometry of L 0, Canad. J . Math . 5 9 No . 5 (2007) , 1029-1049 .

[KKZ] A . Koldobsky , H . Köni g an d M . Zymonopoulou , The complex Busemann-Petty problem on sections of convex bodies, preprint .

[Kl] B . Klartag, On convex perturbations with a bounded isotropic constant, Geom . Funct . Anal. 1 6 (2006) , no . 6 , 1274-1290 .

[KMP] H . König , M . Meyer , A . Pajor , The isotropy constants of the Schatten classes are bounded, Math . Ann . 31 2 (1998) , no . 4 , 773-783 .

[KPY] A . Koldobsky , A . Pajo r an d V . Yaskin , Inequalities of the Kahane-Khinchin type and sections of L p-balls, preprint .

[KRZ] A . Koldobsky , D . Ryabogi n an d A . Zvavitch , Projections of convex bodies and the Fourier transform, Israe l J . Math . 13 9 (2004) , 361-380 .

[KS] A . Koldobsky , C . Shane , The determination of convex bodies from derivatives of section functions, Arch . Math . 8 8 (2007) , 279-288 .

[Kw] S . Kwapien , Problem 3, Studia Math . 3 8 (1970) , 469 . [KYY] A . Koldobsky , V . Yaski n an d M . Yaskina , Modified Busemann-Petty problem on

sections on convex bodies, Israel J . Math . 15 4 (2006) , 191-208 . [KZ] A . Koldobsky an d M . Zymonopoulou , Extremal sections of complex l p-balls, 0 < p <

2, Studi a Math . 15 9 (2003) , 185-194 . [La] P . S . Laplace , Theorie analytique des probabilites, Paris , 1812 . [Lev] P . Levy , Theorie de Vaddition de variable aleatoires, Gauthier-Villars , Paris , 1937 . [Lew] D . R . Lewis , Finite dimensional subspaces of Lp, Studi a Math . 6 3 (1978) , 207-212 . [Li] J . Lindenstrauss , On the extension of Operators with finite dimensional ränge, Illinoi s

J. Math . 8 (1964) , 488-499 . [LR] D . G . Larma n an d C . A . Rogers , The existence of a centrally Symmetrie convex body

with central sections that are unexpectedly small, Mathematik a 2 2 (1975) , 164-175 . [Lu] E . Lutwak , Intersection bodies and dual mixed volumes, Adv . Math . 7 1 (1988) , 232 -

261. [LYZ] E . Lutwak , D . Yang , an d G . Zhang , L p John ellipsoids, Proc . Londo n Math . Soc .

(3) 9 0 (2005) , no . 2 , 497-520 . [Ma] B . Maurey , Theoremes de factorisation pour les Operateurs lineaires ä valeurs dans

les espaces L p, Asterisqu e 11 , 1974 . [Mil] E . Milman , Dual mixed volumes and the slicing problem, Adv . Math . 20 7 (2006) ,

no. 2 , 566-598 . [Mi2] E . Milman , Generalized intersection bodies, J . Funct . Anal . 24 0 (2006) , 530-567 . [Mi3] E . Milman , Generalized intersection bodies are not equivalent, preprint , arXiv :

math/0701779. [MejP] D . Mejfa , Ch . Pommerenke , On spherically convex Univalent funetions, Michiga n

Math. J . 4 7 (2000) , 163-172 . [Mey] M . Meyer , On a problem of Busemann and Petty, unpublishe d manuscript . [MeyP] M . Meyer , A . Pajor , Sections of the unit ball of l™, J . Funct . Anal . 8 0 (1988) ,

109-123. [MiP] V . D . Milman , A . Pajor , Isotropic position and inertia ellipsoids and zonoids of

the unit ball of a normed n-dimensional space, in : Geometri e Aspect s o f Functiona l Analysis, ed. by J.Lindestrauss an d V.D.Milman, Lectur e Notes in Mathematics 1376 , Springer, Heidelberg , 1989 , 64-104 .

104 BIBLIOGRAPHY

[Ne] A . Neyman, Representation of Lp-norms and isometric embedding in L v-spaces, Israe l J. Math . 4 8 (1984) , 129-138 .

[Ni] E . M . Nikishin , Resonance theorems and superlinear Operators, Uspehi Mat . Nau k 2 5 (1970), 129-191 .

[NP] F . L . Nazaro v an d A . N . Podkorytov , Ball, Haagerup, and distribution functions, Complex analysis , Operators , an d relate d topics , Oper . Theor y Adv . Appl . 113 , Birkhäuser, Basel , 2000 , 247-267 .

[O] K . Oleszkiewiez , On p-pseudostable random variables, Rosenthal Spaces and l™ ball slicing, Geometri e aspect s o f funetiona l analysis , Lectur e Note s i n Math . 1807 , Springer-Verlag, Berlin-Heidelberg-Ne w York , 2003 , 188-210 .

[OP] K . Oleszkiewie z an d A . Pelezynski, Polydisc slicing in C n , Studi a Math . 14 2 (2000) , 281-294.

[Pap] M . Papadimitrakis , On the Busemann-Petty problem about the convex, centrally Symmetrie bodies in R n, Mathematik a 3 9 (1992) , 258-266 .

[Pe] C . M . Petty , Projection bodies, Proc . Coli . Convexit y (Copenhage n 1965) , Koben -havns Univ . Mat . Inst. , 234-241 .

[Pog] A . V . Pogorelov , Extrinsic geometry of convex surfaces, Translation s o f Mathemat -ical Monographs , vol . 35 , American Mathematica l Society , Providence , R.I. , 1973 .

[Pol] G . Polya , Berechnung eines bestimmten integrals, Math . Ann . 7 4 (1913) , 204-212 . [Ra] J . G . Ratcliffe , Foundations of hyperbolic manifolds, Springer-Verlag , Ne w York ,

1994. [Ru] B . Rubin , The lower dimensional Busemann-Petty problem for bodies with the gen-

eralized axial symmetry, preprint , arXiv:math/0701317v2 . [RZ] B . Rubi n an d Gaoyon g Zhang , Generalizations of the Busemann-Petty problem for

sections of convex bodies , J . Funct . Anal . 11 8 (1996) , 319-340 . [Schi] J . Schlieper , A note on k-intersection bodies, Proc. Amer . Math . Soc . 13 5 (2007) ,

2081-2088. [Schnl] R . Schneider , Zu einem Problem von Shephard über die Projektionen konvexer

Körper, Math . Zeit . 10 1 (1967) , 71-82 . [Schn2] R . Schneider , Zonoids whose polars are zonoids, Proc . Amer . Math . Soc . 5 0

(1975), 365-368 . [Schn3] R . Schneider , Convex Bodies: the Brunn-Minkowski Theory, Cambridg e Univer -

sity Press , Cambridge , 1993 . [Scho] I . J . Schoenberg , Metrie Spaces and positive definite functions, Trans . Amer . Math .

Soc, 4 4 (1938) , 522-536 . [Se] V . I . Semyanistyi , Some integral transformations and integral geometry in an elliptic

space, Trud y Sem . Vector . Tenzor . Anal . 1 2 (1963) , 397-411 . [Sh] G . C . Shephard , Shadow Systems of convex bodies, Israel J . Math . 2 (1964) , 229-306 . [V] J . D . Vaaler, A geometric inequality with applications to linear forms, Pacifi c J . Math .

83 (1979) , 543-553 . [W] W . Weil , Zonoide und verwandte Klassen konvexer Körper, Monatsh . Math . 9 4

(1982), 73-84 . [Yl] V . Yaskin , The Busemann-Petty problem in hyperbolic and spherical Spaces, Adv .

Math. 20 3 (2006) , 537-553 . [Y2] V . Yaskin , A Solution to the lower dimensional Busemann-Petty problem in the hy-

perbolic space, J . Geom . Anal . 1 6 No . 4 (2006) , 735-745 . [Y3] V . Yaskin , On strict inclusions in hierarchies of convex bodies, preprint ,

arXiv:0707.1471 [math.MG] . [Ya] M . Yaskina , Non-intersection bodies, all of whose central sections are intersection

bodies, Proc. Amer . Math . Soc . 13 5 (2007) , no . 3 , 851-860 . [Zhl] Gaoyon g Zhang , Sections of convex bodies, Amer. J . Math . 11 8 (1996) , 319-340 . [Zh2] Gaoyon g Zhang , A positive answer to the Busemann-Petty problem in four dimen-

sions, Annai s o f Math . 14 9 (1999) , 535-543 .

BIBLIOGRAPHY 10 5

[Zv] A . Zvavitch , The Busemann-Petty problem for arbitrary measures, Math . Ann . 33 1 (2005), 867-887 .

This page intentionally left blank

Index

£ £ , uni t bal l o f C , 2 J3£, uni t b a l l o f ^ , 4 R, spherica l Rado n transform , 1 2 Rm, dua l Rado n transform , 3 4 Rn-k, (n—fc)-dimensiona l spherica l Rado n

transform , 33 üftg, rea l par t o f g , 1 5 \q], ceilin g function , 4 3 TZ, Rado n transform , 1 2 /c-intersection body , 2 4 /c-intersection bod y o f a sta r body , 2 3

Busemann-Petty problem , 3 9 Busemann-Petty proble m for arbitrary mea -

sures, 4 2

Cauchy projectio n formula , 6 0 completely monotoni c function , 5 constant o f isotropy , 9 6 curvature function , 6 0

distribution, 9

embedding i n L p , 6 7

Fourier transfor m o f a distribution , 9 fractional derivative , 1 5 fractional powe r o f th e Laplacian , 4 3

generalized /c-intersectio n body , 3 3

h-convex body , 4 7 hyperbolic space , 4 6

intersection body , 2 1 intersection bod y o f a sta r body , 2 1 isomorphic Busemann-Petty problem , 9 8 isotropic body , 9 6

Levy representation , 6 7 lower dimensional Busemann-Petty prob -

lem, 5 5

Lutwak connections , 3 9

Minkowski functional , 1 1 Minkowski's first inequality , 6 4 Minkowski's uniquenes s theorem , 1 4 mixed volume , 6 4

non-Euclidean Busemann-Pett y problem , 45

normalized positiv e g-stable random vari -able, 7 2

parallel sectio n function , 1 4 Parseval's formula , 1 9 polar projectio n body , 6 8 positive definit e distribution , 1 8 projection body , 6 2

radial metric , 2 1

s-convex body , 4 6 Shephard's problem , 6 3 slicing problem , 9 6 spherical Parseval' s formula , 1 9 spherical space , 4 6 star body , 1 1 support function , 6 1 surface are a measure , 5 9 Symmetrie normalize d g-stabl e rando m

variable, 7 1

test function , 9 totally geodesi c submanifold , 4 7

107

This page intentionally left blank

Titles i n Thi s Serie s

108 Alexande r Koldobsk y an d Vladysla v Yaskin , Th e interfac e betwee n conve x geometr y and harmoni c analysis , 200 8

107 Fa n Chun g an d Linyua n Lu , Comple x graph s an d networks , 200 6

106 Terenc e Tao , Nonlinea r dispersiv e equations : Loca l an d globa l analysis , 200 6

105 Christop h Thiele , Wav e packe t analysis , 200 6

104 Donal d G . Saari , Collisions , rings , an d othe r Newtonia n iV-bod y problems , 200 5

103 Iai n Raeburn , Grap h algebras , 200 5

102 Ke n Ono , Th e we b o f modularity : Arithmeti c o f th e coefficient s o f modula r form s an d q

series, 200 4

101 Henr i Darmon , Rationa l point s o n modula r ellipti c curves , 200 4

100 Alexande r Volberg , Calderön-Zygmun d capacitie s an d Operator s o n nonhomogeneou s

Spaces, 200 3

99 Alai n Lascoux , Symmetri e funetion s an d combinatoria l Operator s o n polynomials , 200 3

98 Alexande r Varchenko , Specia l funetions , K Z typ e equations , an d representatio n theory ,

2003

97 Bern d Sturmfels , Solvin g System s o f polynomia l equations , 200 2

96 N ik y Kamran , Selecte d topic s i n th e geometrica l stud y o f differentia l equations , 200 2

95 Benjami n Weiss , Singl e orbi t dynamics , 200 0

94 Davi d J . Sa l tman , Lecture s o n divisio n algebras , 199 9

93 Gor o Shimura , Eule r produet s an d Eisenstei n series , 199 7

92 Fa n R . K . Chung , Spectra l grap h theory , 199 7

91 J . P . Ma y e t a l . , Equivarian t homotop y an d cohomolog y theory , dedicate d t o th e

memory o f Rober t J . Piacenza , 199 6

90 Joh n Roe , Inde x theory , coars e geometry , an d topolog y o f manifolds , 199 6

89 Cliffor d Henr y Taubes , Metrics , connection s an d gluin g theorems , 199 6

88 Crai g Huneke , Tigh t closur e an d it s applications , 199 6

87 Joh n Eri k Fornaess , Dynamic s i n severa l comple x variables , 199 6

86 Sori n Popa , Classificatio n o f subfactor s an d thei r endomorphisms , 199 5

85 Michi o J imb o an d Tetsuj i Miwa , Algebrai c analysi s o f solvabl e lattic e modeis , 199 4

84 H u g h L . Montgomery , Te n lecture s o n th e interfac e betwee n analyti c numbe r theor y an d

harmonic analysis , 199 4

83 Carlo s E . Kenig , Harmoni c analysi s technique s fo r secon d orde r ellipti c boundar y valu e

problems, 199 4

82 Susa n Montgomery , Hop f algebra s an d thei r action s o n rings , 199 3

81 S teve n G . Krantz , Geometri e analysi s an d funetio n Spaces , 199 3

80 Vaugha n F . R . Jones , Subfactor s an d knots , 199 1

79 Michae l Frazier , Björ n Jawerth , an d Guid o Weiss , Littlewood-Pale y theor y an d th e

study o f funetio n Spaces , 199 1

78 Edwar d Formanek , Th e polynomia l identitie s an d variant s o f n x n matrices , 199 1

77 Michae l Christ , Lecture s o n singula r integra l Operators , 199 0

76 Klau s Schmidt , Algebrai c idea s i n ergodi c theory , 199 0

75 F . Thoma s Farrel l an d L . Edwi n Jones , Classica l aspherica l manifolds , 199 0

74 Lawrenc e C . Evans , Wea k convergenc e method s fo r nonlinea r partia l differentia l

equations, 199 0

73 Walte r A . Strauss , Nonlinea r wav e equations , 198 9

72 Pete r Orlik , Introductio n t o arrangements , 198 9

71 Harr y D y m , J contractiv e matri x funetions , reproducin g kerne l Huber t space s an d interpolation, 198 9

TITLES I N THI S SERIE S

70 Richar d F . Gundy , Som e topic s i n probabilit y an d analysis , 198 9

69 Fran k D . Grosshans , Gian-Carl o Rota , an d Joe l A . Stein , Invarian t theor y an d

superalgebras, 198 7

68 J . Wil l ia m Heiton , Josep h A . Ball , Charle s R . Johnson , an d Joh n N . Palmer ,

Operator theory , analyti c functions , matrices , an d electrica l engineering , 198 7

67 Haral d Upmeier , Jorda n algebra s i n analysis , Operato r theory , an d quantu m mechanics ,

1987

66 G . Andrews , g-Series : Thei r developmen t an d applicatio n i n analysis , numbe r theory ,

combinatorics, physic s an d Compute r algebra , 198 6

65 Pau l H . Rabinowitz , Minima x method s i n critica l poin t theor y wit h application s t o

differential equations , 198 6

64 Donal d S . Passman , Grou p rings , crosse d product s an d Galoi s theory , 198 6

63 Walte r Rudin , Ne w construction s o f function s holomorphi c i n th e uni t bal l o f C n , 198 6

62 Bel a Bol lobäs , Extrema l grap h theor y wit h emphasi s o n probabilisti c methods , 198 6

61 Mogen s Flensted-Jensen , Analysi s o n non-Riernannia n Symmetri e Spaces , 198 6

60 Gille s Pisier , Factorizatio n o f linea r Operator s an d geometr y o f Banac h Spaces , 198 6

59 Roge r How e an d Al le n Moy , Harish-Chandr a homomorphism s fo r p-adi c groups , 198 5

58 H . Blain e Lawson , Jr. , Th e theor y o f gaug e fields i n fou r dimensions , 198 5

57 Jerr y L . Kazdan , Prescribin g th e curvatur e o f a Riemannia n manifold , 198 5

56 Har i Bercovici , Cipria n Foia§ , an d Car l Pearcy , Dua l algebra s wit h application s t o

invariant subspace s an d dilatio n theory , 198 5

55 Wil l ia m Arveson , Te n lecture s o n Operato r algebras , 198 4

54 Wil l ia m Fulton , Introductio n t o intersectio n theor y i n algebrai c geometry , 198 4

53 Wi lhe l m Klingenberg , Close d geodesic s o n Riemannia n manifolds , 198 3

52 Tsi t -Yue n Lam , Orderings , valuation s an d quadrati c forms , 198 3

51 Masamich i Takesaki , Structur e o f factor s an d automorphis m groups , 198 3

50 Jame s Eell s an d Lu c Lemaire , Selecte d topic s i n harmoni c maps , 198 3

49 Joh n M . Franks , Homolog y an d dynamica l Systems , 198 2

48 W . Stephe n Wilson , Brown-Peterso n homology : a n introductio n an d sampler , 198 2

47 Jac k K . Haie , Topic s i n dynami c bifurcatio n theory , 198 1

46 Edwar d G . Effros , Dimension s an d C*-algebras , 198 1

45 Ronal d L . Graham , Rudiment s o f Ramse y theory , 198 1

44 Phil l i p A . Griffiths , A n introductio n t o th e theor y o f specia l divisor s o n algebrai c curves ,

1980

43 Wil l ia m Jaco , Lecture s o n three-manifol d topology , 198 0

42 Jea n Dieudonne , Specia l function s an d linea r representation s o f Li e groups , 198 0

41 D . J . N e w m a n , Approximatio n wit h rationa l functions , 197 9

40 Jea n Mawhin , Topologica l degre e method s i n nonlinea r boundar y valu e problems , 197 9

39 Georg e Lusztig , Representation s o f finite Chevalle y groups , 197 8

38 Charle s Conley , Isolate d invarian t set s an d th e Mors e index , 197 8

37 Masayosh i Nagata , Polynomia l ring s an d affin e Spaces , 197 8

36 Car l M . Pearcy , Som e recen t development s i n Operato r theory , 197 8

35 R . Bowen , O n Axio m A diffeomorphisms , 197 8

34 L . Auslander , Lectur e note s o n nil-thet a functions , 197 7

For a complet e lis t o f t i t le s i n thi s series , visi t t h e AMS Bookstor e a t w w w . a m s . o r g / b o o k s t o r e / .