Flat simplices and kissing polytopes

We consider how flat a lattice simplex contained in the hypercube $[0,k{]}^d$ can be. This question is related to the notion of kissing polytopes: two lattice polytopes contained in the hypercube $[0,k{]}^d$ are kissing when they are disjoint but their distance is as small as possible. We show that the smallest possible distance of a lattice point $P$ contained in the cube $[0,k{]}^3$ to a lattice triangle in the same cube that does not contain $P$ is $$\frac{1}{\sqrt{3k^4-4k^3+4k^2-2k+1}}$$ when $k$ is at least $2$. We also improve the known lower bounds on the distance of kissing polytopes for $d$ at least $4$ and $k$ at least $2$.