Precise asymptotics for expected Hausdorff distance of random polytopes under interior sampling

Determine the precise asymptotic behavior, including the exact leading constant, of the expected Hausdorff distance E[δ_H(Q,Q_n)] between a convex polytope Q ⊂ R^d and the random polytope Q_n formed by the convex hull of n independent points sampled uniformly from the interior of Q, as n → ∞. In particular, refine the known rate E[δ_H(Q,Q_n)] ≍ n^{-1/d} to an exact asymptotic formula that identifies the limiting constant depending on Q.

Background

The paper studies random approximation of convex bodies in the Hausdorff metric, with a focus on expected approximation error when the approximating random polytope is formed by the convex hull of n i.i.d. points sampled uniformly from either the boundary or the interior of the body. While extensive results exist for symmetric difference (volume) metrics, the Hausdorff case is comparatively less developed.

For polytopes, a classical result of I. Barany establishes the order of magnitude E[δ_H(Q,Q_n)] ≍ n^{-1/d} when points are sampled uniformly inside the polytope, but does not provide an exact asymptotic constant or formula. The authors explicitly state that beyond this rate, the precise asymptotic behavior is not known. Their work determines exact constants in two dimensions for boundary sampling on polygons, but the interior-sampling case in general dimension remains unresolved.