Random simplices with small circumradius in a convex window: shape law and window effect

For d + 1 independent uniform points in a convex body P ⊂ Rd with circumradius R ≤ R⋆, we expand the conditional mean of every bounded shape functional to first order in R⋆, with explicit remainder: the limit is the typical Poisson-Delaunay mean, the first-order term a mean-width covariance times the surface-to-volume ratio. For planar convex windows with piecewise C2 boundary the second-order term for the triangle-to-circumdisk area ratio depends only on area, perimeter and exterior angles, and is negative. Simplex windows of every dimension have exact formulas at every cutoff, and the equilateral triangle has mean area ratio 0.152 in closed form. In the unit disk we give the circumradius law on (0,1], joining the known law on [1,∞), and its area-ratio weighting on (0, ∞), with mean area ratio 105/(16π2) − 1/2. The Supplement proves small-radius overlap polynomials for simple polytopes and second-order terms for balls in every dimension, the latter conjectured for smooth bodies.