Given L a convex body, the $L_p$-Busemann Random Simplex Inequality is closely related to the centroid body ${\mathrm{\Gamma }}_{p}L$ 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.

给定L是一个凸体，则${升}_{磷}$-Busemann Random Simplex Inequality 与质心体密切相关 ${\mathrm{\Gamma }}_{磷}升$ 为了 $磷=1$ 和 2，并且只有在这些情况下才可以使用 ${升}_{磷}$-Busemann-Petty 质心不等式。我们定义一个凸体$N_{磷}升$ 并证明等周不等式 ${(N_{磷}升)}^{\circ }$ 这相当于 ${升}_{磷}$-Busemann 随机单纯形不等式。作为应用，我们给出了 Busemann 随机单纯形不等式的一般泛函版本的简单证明，并研究了与佩蒂猜想不等式相关的对偶理论。更准确地说，我们证明了${升}_{磷}$-Busemann 随机单纯形不等式通过p仿射表面积测量，我们证明了 Petty 猜想等价于一个集合和函数${升}_1$-Sharp Affine Sobolev 型不等式强于（并直接暗示）Sobolev-Zhang 不等式。