  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 $K^\circ$.
