A remark on projection outer volume ratio

We study the projection outer volume ratio ${\mathrm{p}\mathrm{o}\mathrm{v}\mathrm{r}}_k(K)$ introduced by D. Galicer, A.E. Litvak, M. Merzbacher, and D. Pinasco. When $k\le n\mathrm{/}2$, we provide sharp bounds in the general case, namely $${\mathrm{p}\mathrm{o}\mathrm{v}\mathrm{r}}_k(K)\le C\max \{1,\sqrt{\frac{k}{\ln (2n\mathrm{/}k)}}\},$$ where the maximal order is obtained for $K=B_1^n$. Moreover, in the important particular case $K=B_2^n$ we show that ${\mathrm{p}\mathrm{o}\mathrm{v}\mathrm{r}}_k(B_2^n)$ is bounded by an absolute constant whenever $k\le n\mathrm{/}2$ and $${\mathrm{p}\mathrm{o}\mathrm{v}\mathrm{r}}_k(B_2^n)\le C\sqrt{\frac{n}{n-k}}\ln (1+\frac{n}{n-k})$$ for $k>n\mathrm{/}2$. The latter bound is sharp up to a logarithmic term.