A note on the perimeter of fat objects

In this Note, 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)≤cdiam(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.

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