2025/03/25 by Giannopoulos, Apostolos, Tziotziou, Natalia · 2 citations
#52A22 #52A23 #62H05 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Primary 60D05 #Probability (math.PR) #Secondary 60E15
paper · doi:10.48550/arxiv.2503.19528
Let μ be a centered log-concave probability measure on \mathbb Rn and let Λμ∗ denote the Cramér transform of μ, i.e. Λμ∗(x)=sup\⟨ x,ξ⟩-Λμ(ξ):ξ∈ℝn\ where Λμ is the logarithmic Laplace transform of μ. We show that 𝔼μ[exp((c1)/(n)Λμ∗ )]<∞ where c1>0 is an absolute constant. In, particular, Λμ∗ has finite moments of all orders. The proof, which is based on the comparison of certain families of convex bodies associated with μ, implies that ‖Λμ∗‖L2(μ)\leqslant c2nln n. The example of the uniform measure on the Euclidean ball shows that this estimate is optimal with respect to n as the dimension n grows to infinity.