5 problems

Conjecture on the admissible range of $nu$. It should be possible to enlarge the range of possible values of the parameter $nu$ to $nu>-2$.

Let $K\subset\mathbb{R}^d$ be a convex body, and let $\bm X_n$ be a set of $n$ independent uniformly chosen random points from $K$. For $k=1,\dots,d-1$, write $\deg_k(\bm X_n)$ for…

Let $B_3^+$ be the three-dimensional halfball described above, and let $o$ be the origin in the center of its base. Consider the expected volume of a random tetrahedron in $B_3^+$…

Let $K$ be a convex body in $\mathbb{R}^d$, let $T^d$ be a $d$-dimensional simplex, let $p\geqslant 1$, and let $n\geqslant d+1$. Write $\mathbb{E}^p_*(K)$ and $\mathbb{E}^p_n(K)$…

Let $K$ be a convex body in $\mathbb{R}^d$ with centroid at $o$, and let $\mathbb{E}^1_o(K)$ denote the normalized expected volume of a random simplex in $K$ whose one vertex is fi…