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 $K^\circ$.

💡 Research Summary

The paper presents a comprehensive extension of the theory of $L_p$ affine geometry to the full range $p\in