8 problems

Covering product conjecture. One has

Let $\omega=(\omega_0,\dots,\omega_d)\in\mathbb{R}^{d+1}_{>0}$ satisfy $\omega_0\leqslant\dots\leqslant\omega_d$, and define the weighted terminal simplex … For $i\in[d]$, Codenott…

Let $T_d=\operatorname{conv}(-\mathbb{1}_d,e_1,\dots,e_d)$ be the terminal simplex in dimension $d$, and let $i\in[d]$. Merino–Schymura conjecture. For every $d\in\mathbb{N}$ and e…

Let $\omega\in\mathbb{R}^{d+1}_{>0}$ satisfy $\omega_0\leq\omega_1\leq\ldots\leq\omega_d$, and let $j\in\{1,\ldots,d\}$. Let $S(\omega)$ be the weighted simplex and let $\sigma_r$…

Let $S(\mathbf{1}_{n+1})$ be the standard terminal simplex, and let $\mu_d$ denote the $d$-th covering minimum. The covering-minima conjecture. For every $n \in \mathbb{N}$ and…

Let … be the simplex, and let $\mu_i(T_n)$ be its $i$th covering minimum. Simplex covering-minima conjecture. The inequalities suggested by the coordinate-subspace restrictions are…

For a convex body $K\in\mathcal{K}^n$, let $\mu_i(K)$ denote its $i$th covering minimum. Covering-product lower-bound conjecture. For every $K\in\mathcal{K}^n$, … with equality, fo…

Let $K\in\mathcal{K}^n$ be a convex body and let $\Lambda\in\mathcal{L}^n$ be a lattice. Let $\mu_1(K,\Lambda)$ denote the first covering minimum. Makai Jr.'s conjecture. One has ……