Hadwiger's problem for bodies with enough sub-Gaussian marginals

Hadwiger’s conjecture in convex geometry, formulated in 1957, states that every convex body in ${\mathbb{R}}^n$ can be covered by $2^n$ translations of its interior. Despite significant efforts, the best known bound related to this problem was $\mathcal{O}(4^n\sqrt{n}\log n)$ for more than sixty years. In 2021, Huang, Slomka, Tkocz, and Vritsiou made a major breakthrough by improving the estimate by a factor of $\exp (\mathrm{\Omega }(\sqrt{n}))$. Further, for ${\psi }_2$ bodies they proved that at most $\exp (-\mathrm{\Omega }(n))\cdot 4^n$ translations of its interior are needed to cover it. Through a probabilistic approach we show that the bound $\exp (-\mathrm{\Omega }(n))\cdot 4^n$ can be obtained for convex bodies with sufficiently many well-behaved sub-gaussian marginals. Using a small diameter approximation, we present how the currently best known bound for the general case, due to Campos, Van Hintum, Morris, and Tiba can also be deduced from our results.