Random Section and Random Simplex Inequality

Let K ⊂ \({\mathbb{R}}^{d}\) ${\mathbb{R}}^d$ be a convex body. Let X₁, . . . , X_k, where k ≤ d, be random points independently and uniformly chosen in K, and let ξ_k be a uniformly distributed random linear k-plane. It is proved that if p ≥ −d + k + 1, then

\({\mathbb{E}}{\left|K\cap {\xi }_{k}\right|}^{d+p}\le {c}_{d,k,p}\cdot {\left|K\right|}^{k}{\mathbb{E}}{\left|{\text{conv}}\left(0,{X}_{1},\dots ,{X}_{k}\right)\right|}^{p},\) $\mathbb{E}{\left|K,\cap ,{\xi }_k\right|}^{d+p}\le c_{d,k,p}·{\left|K\right|}^k\mathbb{E}{\left|\text{conv},\left(0,,,X_1,,,\cdots ,,,X_k\right)\right|}^p,$

where | · | and conv denote the volume of the corresponding dimension and the convex hull, respectively. The constant c_d,k,p is such that if k > 1, then the equality holds if and only if K is an ellipsoid centered at the origin, and if k = 1, then the inequality turns to equality.

The inequality reduces to the Busemann intersection inequality if p = 0 and to the Busemann random simplex inequality if k = d.

An affine version of this inequality which in a similar way generalizes the Schneider inequality and the Blaschke-Grömer inequality is also presented.