Inequalities Relating to L_p-Version of the Petty’s Conjectured Projection Inequality

Petty's conjectured projection inequality is a famous open problem in convex bodies theory.It was shown that an inequality relating to L_p-version of the Petty's conjectured projection inequality by using the notions of the L_p-mixed volume and the L_p-dual mixed volume,the relation of the L_p-projection body and the geometric body Г-_pK,the Bourgain-Milman inequality and the L_p-Busemann-Petty inequality.In addition,for each origin-symmetric convex body,applying the Jensen inequality and the monotonicity of the geometric body Г-_pK,the reverses of L_p-version of the Petty's conjectured projection inequality and the L_p-Petty projection inequality were given respectively.