In geometry, a fat object is an object in two or more dimensions, whose lengths in the different dimensions are similar. For example, a square is fat because its length and width are identical. A 2-by-1 rectangle is thinner than a square, but it is fat relative to a 10-by-1 rectangle. Similarly, a circle is fatter than a 1-by-10 ellipse and an equilateral triangle is fatter than a very obtuse triangle. Fat objects are especially important in computational geometry. Many algorithms in computational geometry can perform much better if their input consists of only fat objects; see the applications section below. Global fatness Given a constant R≥1, an object o is called R-fat if its "slimness factor" is at most R. The "slimness factor" has different definitions in different papers. A common definition is: where o and the cubes are d-dimensional. A 2-dimensional cube is a square, so the slimness factor of a square is 1 (since its smallest enclosing square is the same as its largest enclosed disk). The slimness factor of a 10-by-1 rectangle is 10. The slimness factor of a circle is √2. Hence, by this definition, a square is 1-fat but a disk and a 10×1 rectangle are not 1-fat. A square is also 2-fat (since its slimness factor is less than 2), 3-fat, etc. A disk is also 2-fat (and also 3-fat etc.), but a 10×1 rectangle is not 2-fat. Every shape is ∞-fat, since by definition the slimness factor is always at most ∞. The above definition can be termed two-cubes fatness since it is based on the ratio between the side-lengths of two cubes. Similarly, it is possible to define two-balls fatness, in which a d-dimensional ball is used instead. A 2-dimensional ball is a disk. According to this alternative definition, a disk is 1-fat but a square is not 1-fat, since its two-balls-slimness is √2. An alternative definition, that can be termed enclosing-ball fatness (also called "thickness") is based on the following slimness factor: The exponent 1/d makes this definition a ratio of two lengths, so that it is comparable to the two-balls-fatness. Here, too, a cube can be used instead of a ball. Similarly it is possible to define the enclosed-ball fatness based on the following slimness factor: Enclosing-fatness vs. enclosed-fatness The enclosing-ball/cube-slimness might be very different from the enclosed-ball/cube-slimness. For example, consider a lollipop with a candy in the shape