2019/07/03 by Fodor, Ferenc, Papvári, Dániel I., Vígh, Viktor · 1 citation
#52A22 (Primary) #52A27 #60D05 (Secondary) #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR)
paper · doi:10.48550/arxiv.1907.01868
For two convex discs K and L, we say that K is L-convex 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 C2+ smooth convex discs such that K is L-convex. Select n i.i.d. uniform random points x1,…, xn from K, and consider the intersection K(n) of all translates of L that contain all of x1,…, xn. The set K(n) is a random L-convex polygon in K. We study the expectation of the number of vertices f0(K(n)) and the missed area A(K∖ Kn) 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). 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 (Fodor, Kevei and Vígh (2014)). 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, Kevei and Vígh (2014).