I have a question linked to Interplay of Hausdorff metric and Lebesgue measure. Let us denote as $\mathcal K(\mathbb{R}^n)$ the space of compact subsets of $\mathbb R^n$ endowed with the Hausdorff metric $\rho$ and let $\lambda$ be the $n$-dimensional Lebesgue measure on $\mathbb R^n$. I want to know if there are (sufficient) conditions under which the measure $\lambda$ is continuous w.r.t. $\rho$, that is $$ \lim_{k\rightarrow\infty}\rho(K, K_k)=0\qquad\Rightarrow\qquad \lim_{k\rightarrow\infty}\lambda(K_k)=\lambda(K). $$ I tried to search it in the books Fractal geometry by Kenneth Falconer and Functions of Bounded Variation and Free Discontinuity Problems by Ambrosio, Fusco and Pallara but I did not find anything. In the second book it is written that, in the case $n=2$, the Hausdorff measure (which is a rescaling of the usual $\lambda$ on $\mathbb R^n$) is lower-semicontinuous w.r.t. the Hausdorff metric along sequences satisfying a suitable uniform concentration property, but this is not what I am looking for.
Some help? Do you have some references?
Claim: If the upper Minkowski dimension of $\partial K$ is strictly less than $n$ then $\rho(K,K_k)\to0$ implies $\lambda(K_k)\to \lambda(K)$.
Proof: Let $\epsilon>0$ and choose $k_0$ such that $\rho(K,K_k)<\epsilon$ for all $k>k_0$. Let $D_k$ be the symmetric difference between $K$ and $K_k$. Then $D_k\subset (\partial K)_{\epsilon}$. Let $d$ be the upper Minkowski dimension of $\partial K$, let $d<d'<n$, and let $(B_{i,\epsilon})_{i=1} ^{N}$ be a collection of $N\leq \epsilon^{-d'}$ (*) balls of diameter $\epsilon$ such that $\partial K\subset \bigcup_{i}B_{i,\epsilon}$. Then $$ (\partial K)_{\epsilon}\subset \bigcup_{i}B_{i,2\epsilon} $$ and therefore $$ \lambda(D_k)\leq \lambda((\partial K)_{\epsilon}) \leq N (2\epsilon)^{n}\leq 2^n\epsilon^{n-d'}\to 0 $$
(*) By the definition of the Minkowski dimension, for any $d'>d$ there exists $\epsilon_0>0$ such that $N_{\epsilon}<\epsilon^{-d'}$ is possible for all $\epsilon<\epsilon_0$.