4 problems

In the space $X^n$, let $d_n(r)$, for $n\geq 2$, denote the density of $n+1$ mutually tangent spheres, or horospheres when $r=\infty$, of radius $r$ relative to the simplex spanned…

Makai Jr.'s conjecture. One has

A three-valent Archimedean tiling is a tiling whose vertices have valence three and whose faces are regular polygons arranged in a uniform pattern. Its face-incircles are the circl…

Let $d_n(r)$ denote the density of the regular hyperbolic simplex packing in $n$ dimensions by balls of radius $r$. Monotonicity conjecture. The function $d_n(r)$ is strictly incre…