Bill Taylor wrote: > > firstname.lastname@example.org (Robert Hill) writes: [snip] > |> Another question: is there some interesting shape of bounded set > |> such that the probability is 3/4 for vertices independently and > |> uniformly distributed within that set? > > What an excellent question! I hope someone finds one. Meanwhile we > need those n-gon results... > [snip]
Since the probability is (more or less evidently) less than 3/4 for vertices i.i.d. uniformly in a square and since the probablility approaches 1 as the (1 x 1) square is stretched to become a (1 x L) rectangle (where L --> \infty). If one believes in continuity of the resulting probability as a function of L, then the answer is "Yes, there is a bounded rectangle for which the probability = 3/4." Indeed, such L isn't very big at all.