I am trying to find a constant such that $\lambda_k^{*}(E)<C \cdot\text{diam} (E)^k$ for all $E \subset \mathbb{R}^k$, where $\lambda_k^{*}$ is the outer Lebesgue measure.
I started by assuming $E$ was an interval, ie. $E=(a_1,b_1)\times\cdots\times(a_k,b_k)$ and I think that $\text{diam} (E)=\sqrt{\sum_{i=1}^k(b_i-a_i)^2}$ and $\lambda_k^{*}(E)=\prod_{i=1}^k(b_i-a_i)$. However, I cannot find the desired constant in this case. I'd be grateful if someone could point me in the right direction.
By the AM-GM inequality
$$\lambda^*_k(E) = \prod_{i=1}^k(b_i-a_i) =\left( \prod_{i=1}^k(b_i-a_i)^2 \right)^{\frac{1}{2}} \leq \left(\frac{1}{k} \sum_{i=1}^k(b_i-a_i)^2 \right)^{\frac{k}{2}} = \frac{1}{k^{\frac{k}{2}}} \text{diag}(E)^k$$
Now you just need to select a $C>\frac{1}{k^{\frac{k}{2}}}$.