ON THE $L_p$-VERSION OF THE PETTY'S CONJECTURED PROJECTION INEQUALITY AND APPLICATIONS

Abstract

Petty's conjectured projection inequality is a famous open problem in convex bodies theory. In this paper, it is shown that a $L_p$-version of the Petty's conjectured projection inequality. As its applications, we give a reverse of the Blaschke-Santaló inequality and consider the monotony of volumes for convex body and its $L_p$-Petty projection body, respectively. Otherwise, we also give the reverses of the $L_p$-Petty projection inequality.