Bonnesen inequality

One of the more precise forms of the isoperimetric inequality for convex domains in the plane. Let $K$ be a convex domain in the plane, let $r$ be the radius of the largest circle which can be inserted in $K$, let $R$ be the radius of the smallest circle containing $K$, let $L$ be the perimeter and let $F$ be the area of $K$. The Bonnesen inequality [1]

$$\Delta=L^2-4\pi F\geq\pi^2(R-r)^2$$

is then valid. The equality $\Delta=0$ is attained only if $R=r$, i.e. if $K$ is a disc. For generalizations of the Bonnesen inequality see [2].

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