5 problems

Existence classification conjecture. There exist $\kappa_i=\kappa_i(\mu)\in(0,\pi^2/4)$ with $\kappa_1<\kappa_2<\kappa_3$ such that the curved restricted three-body problem has exa…

Minimal vertex-count conjecture. (i) For a CCP of a topological torus without self-intersection, the minimal number of vertices is $9$. (ii) For an orientable CCP of genus $g$, if…

Let a Segre–Mahler polytope be a convex $n$-dimensional polytope in a space of constant curvature whose every dihedral angle is at most $120^\circ$. Consider a packing of equal sph…

Let $d_n(r)$ denote the density of $n+1$ mutually touching balls of radius $r$ in an $n$-dimensional space of constant curvature, measured with respect to the simplex spanned by th…

The setting is that of polyhedra in constant-curvature three-dimensional model spaces, with tetrahedral faces flatly embedded in the three-sphere $\mathrm{S}^3$ or hyperbolic three…