New $L_p$ Affine Isoperimetric Inequalities

We prove new $L_p$ affine isoperimetric inequalities for all $ p \in [-\infty,1)$\mathrm{.}Weestablish,forall$p\neq -n$, a duality formula which shows that $L_p$ affine surface area of a convex body $K$ equals $L_{\frac{n^2}{p}}$ affine surface area of the polar body $K^{\circ }$.