2023/10/22 by Galicer, Daniel, Singer, Joaquín
#52A20 (secondary) #52A23 #52A38 #52A40 (primary) #52C17 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2310.14381
Hadwiger's conjecture in convex geometry, formulated in 1957, states that every convex body in ℝn can be covered by 2n translations of its interior. Despite significant efforts, the best known bound related to this problem was O(4n √(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(Ω(√(n))). Further, for ψ2 bodies they proved that at most exp(-Ω(n))⋅4n translations of its interior are needed to cover it. Through a probabilistic approach we show that the bound exp(-Ω(n))⋅4n 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.