approximated by: Fatness of a convex polygon A convex polygon is called r-separated if the angle between each pair of edges (not necessarily adjacent) is at least r. Lemma: The enclosing-ball-slimness of an r-separated convex polygon is at most . A convex polygon is called k,r-separated if: It does not have parallel edges, except maybe two horizontal and two vertical. Each non-axis-parallel edge makes an angle of at least r with any other edge, and with the x and y axes. If there are two horizontal edges, then diameter/height is at most k. If there are two vertical edges, then diameter/width is at most k. Lemma: The enclosing-ball-slimness of a k,r-separated convex polygon is at most . improve the upper bound to . Counting fat objects If an object o has diameter 2a, then every ball enclosing o must have radius at least a and volume at least Vdad. Hence, by definition of enclosing-ball-fatness, the volume of an R-fat object with diameter 2a must be at least: Vdad/Rd. Hence: Lemma 1: Let R≥1 and C≥0 be two constants. Consider a collection of non-overlapping d-dimensional objects that are all globally R-fat (i.e. with enclosing-ball-slimness ≤ R). The number of such objects of diameter at least 2a, contained in a ball of radius C⋅a, is at most: For example (taking d=2, R=1 and C=3): The number of non-overlapping disks with radius at least 1 contained in a circle of radius 3 is at most 32=9. (Actually, it is at most 7). If we consider local-fatness instead of global-fatness, we can get a stronger lemma: Lemma 2: Let R≥1 and C≥0 be two constants. Consider a collection of non-overlapping d-dimensional objects that are all locally R-fat (i.e. with local-enclosing-ball-slimness ≤ R). Let o be a single object in that collection with diameter 2a. Then the number of objects in the collection with diameter larger than 2a that lie within distance 2C⋅a from object o is at most: For example (taking d=2, R=1 and C=0): the number of non-overlapping disks with radius larger than 1 that touch a given unit disk is at most 42=16 (this is not a tight bound since in this case it is easy to prove an upper bound of 5). Generalizations The following generalization of fatness were studied by for 2-dimensional objects. A triangle ∆ is a (β, δ)-triangle of a planar object o (0<β≤π/3, 0<δ< 1), if ∆