In article <email@example.com>, firstname.lastname@example.org (Bill Taylor) wrote:
> email@example.com (Robert Hill) writes:
> |> 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...
I thought it was an excellent question too, and sent an answer by email. But it turns out to be a triviality. Consider randomly selecting the vertices from a rectangle of of height 1. The probability of obtuseness is obvioulsy a continuous function of the length of the rectangle. As the square and a long thin rectangle give probabilites either side of 3/4, the intermediate value theorem says there is a rectangle with the given property.
Terry Moore, Statistics Department, Massey University, New Zealand.
Theorems! I need theorems. Give me the theorems and I shall find the proofs easily enough. Bernard Riemann