2 problems

Let $\mathcal N=\{n\in\mathbb N:n\not\equiv 0,4,7\pmod{8}\}$, let $\mathcal E(n)=\{\mathbf{x}\in\mathbb Z^3:|\mathbf{x}|^2=n\}$, let…

Let $K$ be a smooth convex body with volume one in ${\mathbb R}^d$. Let $\Pi_n$ be the Poisson random polytope in $K$ and let $K_n$ be the convex hull of $n$ independent uniformly…