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 \(\exp\left(\Omega(\sqrt{n})\right)\). Further, for \(\psi_2\) bodies they proved that at most \(\exp(-\Omega(n))\cdot4^n\) translations of its interior are needed to cover it. Through a probabilistic approach we show that the bound \(\exp(-\Omega(n))\cdot4^n\) 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.