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. For p ≥ 1, sharp isoperimetric inequalities for the L p -sinetransform of isotropic measures are established. The corresponding re-verse inequalities are obtained in an asymptotically optimal form. Asapplications of our main results, we present volume inequalities for con-vex bodies which are in L p surface isotropic position. 1. IntroductionThe setting for this article is Euclidean n-space R n with n ≥ 3. We use| · | to denote the standard Euclidean norm on R n and we write x · y for thestandard inner product of x,y ∈ R n .A non-negative nite Borel measure µ on the unit sphere S n−1 is said to beisotropic if it has the same moment of inertia about all lines through the originor, equivalently, if for all x ∈ R n ,|x| 2 =Z S n−1 |x ·u| 2 dµ(u).Two basic examples of isotropic measures on S n−1 are (suitably normalized)spherical Lebesgue measure and the cross measures, i.e., measures concentrateduniformly on{±b 1 ,...,±b n }, whereb 1 ,...,b n denote orthonormalbasisvectorsof R

VOLUME INEQUALITIES FOR THE Lp-SINE TRANSFORM OF ISOTROPIC MEASURES

VOLUME INEQUALITIES FOR THE L_p-SINE TRANSFORM OF ISOTROPIC MEASURES