2025/08/21 by Pafis, Minas, Tziotziou, Natalia
#52A23 #60E15 #62H05 #FOS: Mathematics #Metric Geometry (math.MG) #Primary 60D05 #Probability (math.PR) #Secondary 52A22
paper · doi:10.48550/arxiv.2508.15261
Let \Xi\i=1∞ be a sequence of independent copies of a random vector X in ℝn. We revisit the question to determine the asymptotic shape of the random polytope KN=\rm conv\X1,… ,XN\ where N>n. We show that for any β∈ (0,1) there exists a constant c(β)>0 such that the following holds true: If μ is a Borel probability measure on \mathbb Rn then, for all N≥ c(β)n we have that KN⊇ Tβln((N)/(n))(μ) with probability greater than 1-exp(-\tfrac12N1-βnβ), where Tp(μ) is the convex set of all points x∈ℝn with half-space depth greater than or equal to e-p. Our approach does not require any additional assumptions about the measure μ and hence it generalizes and/or improves a sequence of previous results. Moreover, for the class of strongly regular measures we compare the family \Tp(μ)\p>0 to other natural families of convex bodies associated with μ, such as the Lp-centroid bodies of μ or the level sets of the Cramér transform of μ, and use this information in order to estimate the size of a random KN.