Hyperplane conjecture for quotient spaces of $L_p$

Abstract

We give a positive solution for the hyperplane conjecture of quotient spaces F of $L_p$, where $1<p\kll\infty$. \[ vol(B_F)^{\frac{n-1}{n}} \kl c_0 \pl p' \pl \sup_{H \p hyperplane} vol(B_F\cap H) \pl.\] This result is extended to Banach lattices which does not contain ${\ell }_1^n$'s uniformly. Our main tools are tensor products and minimal volume ratio with respect to $L_p$-sections.