about small thin areas that are part of a large fat object. For example, consider a lollipop with a candy in the shape of a 1×1 square and a stick in the shape of a 1×(1/b) rectangle (with b>1>(1/b)). As b increases, the area of the enclosing cube (=4) and the area of the enclosed cube (=1) remain constant, while the total area of the shape changes only slightly (=1+1/b). Thus all three slimness factors are bounded: enclosing-cube-slimness≤2, enclosed-cube-slimness≤2, two-cube-slimness=2. Thus by all definitions the lollipop is 2-fat. However, the stick-part of the lollipop obviously becomes thinner and thinner. In some applications, such thin parts are unacceptable, so local fatness, based on a local slimness factor, may be more appropriate. For every global slimness factor, it is possible to define a local version. For example, for the enclosing-ball-slimness, it is possible to define the local-enclosing-ball slimness factor of an object o by considering the set B of all balls whose center is inside o and whose boundary intersects the boundary of o (i.e. not entirely containing o). The local-enclosing-ball-slimness factor is defined as: The 1/2 is a normalization factor that makes the local-enclosing-ball-slimness of a ball equal to 1. The local-enclosing-ball-slimness of the lollipop-shape described above is dominated by the 1×(1/b) stick, and it goes to ∞ as b grows. Thus by the local definition the above lollipop is not 2-fat. Global vs. local definitions Local-fatness implies global-fatness. Here is a proof sketch for fatness based on enclosing balls. By definition, the volume of the smallest enclosing ball is ≤ the volume of any other enclosing ball. In particular, it is ≤ the volume of any enclosing ball whose center is inside o and whose boundary touches the boundary of o. But every such enclosing ball is in the set B considered by the definition of local-enclosing-ball slimness. Hence: enclosing-ball-slimnessd = = volume(smallest-enclosing-ball)/volume(o) ≤ volume(enclosing-ball-b-in-B)/volume(o) = volume(enclosing-ball-b-in-B)/volume(b ∩ o) ≤ (2 local-enclosing-ball-slimness)d Hence: enclosing-ball-slimness ≤ 2⋅local-enclosing-ball-slimness For a convex body, the opposite is also true: local-fatness implies global-fatness. The proof is based on the following lemma. Let o be a convex object. Let P be a point in o. Let b and B be two balls centered at P such that b is smaller than B. Then o intersects a larger portion of b than of B, i.e.: Proof sketch: standing at the point P, we can look at different