Random section and random simplex inequality

Consider some convex body $K \subset \mathbb{R}^d$. Let $X_1,\dots,X_k$, where $k \le d$, be random points independently and uniformly chosen in $K$, and let $\xi_k$ be a uniformly distributed random linear $k$ - plane. We show that for $p\ge-d+k+1$,
$$ \mathbb{E} |K \cap \xi_k |^{d+p} \le c_{d,k,p} \cdot |K |^k \mathbb{E} | \mathrm{conv}(0,X_1, ..., X_k)|^p ,$$
where $|\cdot|$ and $\mathrm{conv}$ denote the volume of correspondent dimension and the convex hull. The constant $c_{d,k,p}$ is such that for $k>1$ the equality holds if and only if $K$ is an ellipsoid centered at the origin, and for $k=1$ the inequality turns to equality.
If $p=0$, then the inequality reduces to the Busemann intersection inequality, and if $k=d$ — to the Busemann random simplex inequality.
We also present an affine version of this inequality which similarly generalizes the Schneider inequality and the Blaschke-Grömer inequality.
Based on a joint work with Alexander Litvak.