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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 showsthat $L_p$ affine surface area of a convex body $K$ equals $L_\frac{n^2}{p}$affine surface area of the polar body $K^\circ$.

New $L_p$ Affine Isoperimetric Inequalities