  Given $L$ a convex body, the $L_p$-Busemann Random Simplex Inequality is
closely related to the centroid body $\Gamma_p L$ for $p=1$ and $2$, and only
in these cases it can be proved using the $L_p$-Busemann-Petty centroid
inequality. We define a convex body $N_p L$ and prove an isoperimetric
inequality for $(N_p L)^\circ$ that is equivalent to the $L_p$-Busemann Random
Simplex Inequality. As applications, we give a simple proof of a general
functional version of the Busemann Random Simplex Inequality and study a dual
theory related to Petty's conjectured inequality. More precisely, we prove dual
versions of the $L_p$-Busemann Random Simplex Inequality for sets and functions
by means of the $p$-affine surface area measure, and we prove that the Petty
conjecture is equivalent to an $L_1$-Sharp Affine Sobolev-type inequality that
is stronger than (and directly implies) the Sobolev-Zhang inequality.
