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 the 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 several important results , including a n analytic Solution 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 Fourier approac h t o convexity were reflected i n

the book "Fourie r Analysis in Convex Geometry" writte n by the first-name d

author an d publishe d i n 2005 in the Mathematical 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 given at 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