The product-body universality conjecture for random polytopes

Let $m\in\mathbb{N}$. For $i\in\\{1,\dotsc,m\\}$, let $K_i\subset \mathbb{R}^{d_i}$ be a convex body whose boundary is twice differentiable with positive Gaussian curvature everywhere and such that

$$\operatorname{Vol}_{d_i}(K_i)=\operatorname{Vol}_{d_i}(B_2^{d_i}).$$

Set $K=K_1\times\cdots\times K_m\subset \mathbb{R}^d$, where $d=d_1+\cdots+d_m$, and let $\mathcal{P}_{n,d}(K)$, for $n\geq d+1$, be the convex hull of $n$ independent uniformly distributed random points in $K$. Let $Z_d$ denote the corresponding product of Euclidean balls. Product-body universality conjecture. For every $j\in\\{0,\dotsc,d-1\\}$,

$$\mathbb{E} f_j(\mathcal{P}_{n,d}(K))\asymp\mathbb{E} f_j(\mathcal{P}_{n,d}(Z_d)),$$

and

$$\operatorname{Vol}_d(K)-\mathbb{E}\operatorname{Vol}_d(\mathcal{P}_{n,d}(K))\asymp\operatorname{Vol}_d(Z_d)-\mathbb{E}\operatorname{Vol}_d(\mathcal{P}_{n,d}(Z_d)).$$

The conjecture extends the comparison results from products of Euclidean balls to products of general smooth, positively curved convex bodies; it is verified in the paper for products of ellipsoids, but remains open in the stated generality.