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We investigate polytopes inscribed into a sphere that are normally equivalent(or strongly isomorphic) to a given polytope $P$. We show that the associatedspace of polytopes, called the inscribed cone of $P$, is closed under Minkowskiaddition. Inscribed cones are interpreted as type cones of ideal hyperbolicpolytopes and as deformation spaces of Delaunay subdivisions. In particular,testing if there is an inscribed polytope normally equivalent to $P$ ispolynomial time solvable.  Normal equivalence is decided on the level of normal fans and we study thestructure of inscribed cones for various classes of polytopes and fans,including simple, simplicial, and even. We classify (virtually) inscribablefans in dimension $2$ as well as inscribable permutahedra and nestohedra.  A second goal of the paper is to introduce inscribed virtual polytopes.Polytopes with a fixed normal fan $\mathcal{N}$ form a monoid with respect toMinkowski addition and the associated Grothendieck group is called the typespace of $\mathcal{N}$. Elements of the type space correspond to formalMinkowski differences and are naturally equipped with vertices and hence with anotion of inscribability. We show that inscribed virtual polytopes form asubgroup, which can be non-trivial even if $\mathcal{N}$ does not have actualinscribed polytopes.  We relate inscribed virtual polytopes to routed particle trajectories, thatis, piecewise-linear trajectories of particles in a ball with restricteddirections. The state spaces gives rise to connected groupoids generated byreflections, called reflection groupoids. The endomorphism groups of reflectiongroupoids can be thought of as discrete holonomy groups of the trajectories andwe determine when they are reflection groups.

Inscribable fans I: Inscribed cones, virtual polytopes, and routed particle trajectories