mg.metric geometry – Relationship between Lebesgue measure and Hausdorff metric


Let $A, Bsubsetmathcal K(mathbb{R}^d)$ ($=$ compact subsets of $mathbb R^d$) such that $Asubset B$. I would like to know if there are conditions on the sets $A, B$ for which
$$
lambda(Bsetminus A)leq Crho (A, B)quadtext{for some} C>0,
$$

where $lambda$ is the $d$-dimensional Lebesgue measure and $rho$ is the Hausdorff distance on $mathcal K(mathbb R^d)$.