angles θ and measure the distance to the boundary of o. Because o is convex, this distance is a function, say r(θ). We can calculate the left-hand side of the inequality by integrating the following function (multiplied by some determinant function) over all angles: Similarly we can calculate the right-hand side of the inequality by integrating the following function: By checking all 3 possible cases, it is possible to show that always . Thus the integral of f is at least the integral of F, and the lemma follows. The definition of local-enclosing-ball slimness considers all balls that are centered in a point in o and intersect the boundary of o. However, when o is convex, the above lemma allows us to consider, for each point in o, only balls that are maximal in size, i.e., only balls that entirely contain o (and whose boundary intersects the boundary of o). For every such ball b: where is some dimension-dependent constant. The diameter of o is at most the diameter of the smallest ball enclosing o, and the volume of that ball is: . Combining all inequalities gives that for every convex object: local-enclosing-ball-slimness ≤ enclosing-ball-slimness For non-convex objects, this inequality of course doesn't hold, as exemplified by the lollipop above. Examples The following table shows the slimness factor of various shapes based on the different definitions. The two columns of the local definitions are filled with "*" when the shape is convex (in this case, the value of the local slimness equals the value of the corresponding global slimness): Fatness of a triangle Slimness is invariant to scale, so the slimness factor of a triangle (as of any other polygon) can be presented as a function of its angles only. The three ball-based slimness factors can be calculated using well-known trigonometric identities. Enclosed-ball slimness The largest circle contained in a triangle is called its incircle. It is known that: where Δ is the area of a triangle and r is the radius of the incircle. Hence, the enclosed-ball slimness of a triangle is: Enclosing-ball slimness The smallest containing circle for an acute triangle is its circumcircle, while for an obtuse triangle it is the circle having the triangle's longest side as a diameter. It is known that: where again Δ is the area of a triangle and R is the radius of the circumcircle. Hence, for an acute triangle, the