New estimates for $d_{2,1}$ and $d_{3,2}$

Let $K$ be a convex body in ${\mathbb{R}}^n$. Let $d_{n,n-1}(K)$ be the smallest possible density of a non-separable lattice of translates of $K$. In this paper we prove the estimate $d_{2,1}(K)\le \frac{\pi \sqrt{3}}{8}$ for $K\subset {\mathbb{R}}^2$, with equality if and only if $K$ is an ellipse, which was conjectured by E. Makai. Also we prove the estimate $d_{3,2}(K)\le \frac{\pi }{4\sqrt{3}}$ for $K\subset {\mathbb{R}}^3$ using projection bodies.