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Equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points. (English) Zbl 0882.52004

An important general formula is proved, which relates the equiaffine inner parallel curves of a planar convex body \(K\) and the probability \(p_{jk} (K)\) that the convex hull of \(j\) independent random points is disjoint from the convex hull of \(k\) further independent random points. The equiaffine curve \(M_s\) is determined by the midpoints of all chords that divide \(K\) into two parts of areas \(s\) and \(1-s\) (it is assumed that \(K\) has unit area). Let \(w(z, M_s)\) be the winding number of the curve \(M_s\) around the point \(z\).
The authors prove that the number \[ K_{[s]} =1-\int_{z\in K\backslash M_s} w(z,M_s)dz \] is related to \(p_{jk} (K)\) through a remarkable formula \[ p_{jk} (K)= {4\over 3} jk\int^1_0 s^{j-1} (1-s)^{k-1} K_{[s]} ds. \] This relation is used to obtain a number of new results on \(p_{jk} (K)\), which improve an estimate obtained for a special case by L. C. G. Rodgers with the best possible bound. Since \(p_{n-1,1} (K)\) is related to both the expected number of vertices and the expected area of the convex hull of \(n\) uniformly distributed points within \(K\), the obtained result extends the asymptotic formula of Renyi and Sulanke to an asymptotic expansion.

MSC:

52A22 Random convex sets and integral geometry (aspects of convex geometry)
60D05 Geometric probability and stochastic geometry
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