5 problems

Let $d$ and $n$ be as in the beta-polytope model, and fix parameters $\beta_1,\ldots,\beta_n\geq -1$. Write $\mathscr{P}_{n,d}^{\beta_1,\ldots,\beta_n}$ for the corresponding beta…

Floor-probability bounds conjecture. For all $n\geq 3$ and all $K\in\operatorname{CCSF}([0,1])$,

Lower-dimensional volume conjecture. A $d$-dimensional volume for an $m$-dimensional object, where $d<m$, undergoing a fragmentation process follows the Benford distribution for th…

Rectangle-perimeter conjecture. The perimeters of the sub-rectangles from a rectangle undergoing an unrestricted fragmentation process converge to Benford behavior.

Let $K,L\in\mathcal{K}_n$ be convex bodies, and for a unit vector $u\in\mathbb{R}^n$ let $K_u$ and $L_u$ denote their projections onto the hyperplane orthogonal to $u$. Suppose tha…