New $L_p$ Affine Isoperimetric Inequalities

We prove new $L_p$ affine isoperimetric inequalities for all $p \in [-\infty,1)$. We establish, for all $p\neq -n$, 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 $K^\circ$.