On Random Approximations by Generalized Disc‐polygons

For two convex discs K and L, we say that K is L-convex (Lángi et al., Aequationes Math.85(1-2) (2013), 41-67) if it is equal to the intersection of all translates of L that contain K.In L-convexity, the set L plays a similar role as closed half-spaces do in the classical notion of convexity.We study the following probability model: Let K and L be C 2 + smooth convex discs such that K is L-convex.Select n independent and identically distributed uniform random points x 1 , . . ., x n from K, and consider the intersectionWe study the expectation of the number of vertices f 0 (K (n) ) and the missed area A(K \ K n ) as n tends to infinity.We consider two special cases of the model.In the first case, we assume that the maximum of the curvature of the boundary of L is strictly less than 1 and the minimum of the curvature of K is larger than 1.In this setting, the expected number of vertices and missed area behave in a similar way as in the classical convex case and in the r-spindle convex case (when L is a radius r circular disc), see (Fodor et al., Adv. in Appl.Probab.46(4) (2014), 899-918).The other case we study is when K = L.This setting is special in the sense that an interesting phenomenon occurs: the expected number of vertices tends to a finite limit depending only on L. This was previously observed in the special case when L is a circle of radius r in Fodor et al. (Adv. in Appl.Probab.46(4) (2014), 899-918).We also determine the extrema of the limit of the expectation of the number of vertices of L (n) if L is a convex discs of constant width 1.The formulas we prove can be considered as generalizations of the corresponding r-spindle convex statements proved by Fodor et al. in (Adv. in Appl.Probab.46(4) (2014), 899-918).