Is the worst-case translative covering density of convex bodies linear in the dimension?

Translative paper, Theorem 1.1: for absolute $c>0$ and all large $n$ there is a centrally symmetric convex body (even a rational polytope) with $\theta_T(K_n)>cn\log n$, so no universal linear bound exists; with Rogers' bound the worst-case order is $n\log n$. It extends the lattice lower bound of Li and Liu (2026) to arbitrary translative coverings. The companion proves every convex body has a single-lattice covering of density at most $Cn\log n$, removing the $(\log\log n)^{O(1)}$ loss of Li-Liu, so the lattice supremum also has order $n\log n$. Constants are not optimized; Euclidean balls are not treated.

Translative paper, Theorem 1.1: for absolute $c>0$ and all large $n$ there is a centrally symmetric convex body (even a rational polytope) with $\theta_T(K_n)>cn\log n$, so no universal linear bound exists; with Rogers' bound the worst-case order is $n\log n$. It extends the lattice lower bound of Li and Liu (2026) to arbitrary translative coverings. The companion proves every convex body has a single-lattice covering of density at most $Cn\log n$, removing the $(\log\log n)^{O(1)}$ loss of Li-Liu, so the lattice supremum also has order $n\log n$. Constants are not optimized; Euclidean balls are not treated.