Flat simplices and kissing polytopes

Abstract: 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 $2$.