A Sharp Quantitative Estimate for the Perimeters of Convex Sets in the Plane

Let $E \subset B \subset \mathbb R^2$ be bounded, convex sets. The monotonicity of the perimeters holds, i.e. $\mathcal{H}^{1} (\partial E) \leqslant \mathcal{H}^{1}(\partial B)$. Here we give a quantitative estimate of the difference of the perimeters: it shows how much the perimeter of $B$ increases when $B$ becomes larger and larger with respect to $E$. We give an example showing that our estimate is sharp.