A note on the perimeter of fat objects

Abstract

In this Note, we show that the size of the perimeter of (alpha, beta)-covered objects is a linear function of the diameter. Specifically, for an (alpha, beta)-covered object O, per(O) <= c diam(O)/alpha beta sin(2)alpha , 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 gamma-fat objects are a generalization of (alpha, beta)-covered objects. We show that no such relationship between perimeter and diameter can hold for locally gamma-fat objects. Crown Copyright (C) 2010 Published by Elsevier B.V. All rights reserved.