On the perimeter of fat objects

We show that the size of the perimeter of (α,β)-covered objects is a linear function of the diameter. Specifically, for an (α,β)- covered object O, per(O)≤c diam(O)/ αβsin2α, for a positive constant c. One easy consequence of the result is that every point on the boundary of such an object sees a constant fraction of the boundary. Locally γ-fat objects are a generalization of (α,β){covered objects. We show that no such relationship between perimeter and diameter can hold for locally γ-fat objects.