3 Rupert pairs
Write \(\pi \colon \operatorname{\mathbb {R}}^3 \to \operatorname{\mathbb {R}}^2\) for the projection \((x_0, x_1, x_2) \mapsto (x_0, x_1)\) onto the screen.
A convex polyhedron with vertex set \(V\) is Rupert if there are rotations \(R_i, R_o \in SO(3)\) and an offset \(o \in \operatorname{\mathbb {R}}^2\) such that
The main statement: there is a finite set \(V \subseteq \operatorname{\mathbb {R}}^3\) whose convex hull has nonempty interior and which is not Rupert.
3.1 Rupert sets
The convex polyhedron with vertex set \(V\) is Rupert if and only if the convex hull of \(V\) is a Rupert set, i.e. there is a pose (two rotations and an offset) under which the closure of the inner shadow lies in the interior of the outer shadow.
The convex hull of a finite set is compact, so its shadow is closed.
Let \(S \subseteq \operatorname{\mathbb {R}}^3\) be point-symmetric and convex. If a pose with offset \(o\) passes \(S\) through itself, then so does the same pose with offset \(0\).
Write \(K\) for the closure of the inner shadow and \(L\) for the interior of the outer shadow; both are point-symmetric and \(L\) is convex. From \(o + K \subseteq L\) and symmetry, \(-o + K = -(o + K) \subseteq L\). Every \(k \in K\) is the midpoint of \(o + k \in L\) and \(-o + k \in L\), so \(k \in L\) by convexity.
3.2 Shadows and the separation criterion
From now on \(\operatorname{\mathbf{P}}\) denotes the Noperthedron (Definition 2).
The shadow of a \(3 \times 3\) matrix \(R\) is \(\mathrm{Sh}(R) = \pi (R\, \operatorname{\mathbf{P}})\). A pair of matrices \((R_o, R_i)\) is a Rupert pair if \(\overline{\mathrm{Sh}(R_i)} \subseteq \operatorname {int} \mathrm{Sh}(R_o)\).
If \(\operatorname{\mathbf{P}}\) is Rupert, then there is a Rupert pair \((R_o, R_i)\) with \(R_o, R_i \in SO(3)\).
For \(w = (w_0, w_1) \in \operatorname{\mathbb {R}}^2\) write \(\langle w, x \rangle = w_0 x_0 + w_1 x_1\) for \(x \in \operatorname{\mathbb {R}}^3\) (only the screen coordinates enter).
Let \(S\) be a vertex of \(\operatorname{\mathbf{P}}\) and \(w \ne 0\). If
for every vertex \(T\) of \(\operatorname{\mathbf{P}}\), then \((R_o, R_i)\) is not a Rupert pair.
The outer shadow is the convex hull of the points \(\pi (R_o T)\), so it lies in the closed half-plane \(\{ z : \langle w, z \rangle \le \langle w, R_i S \rangle \} \). The point \(\pi (R_i S)\) of the inner shadow lies on the boundary line of this half-plane, so it is not an interior point of the outer shadow.
Only a non-strict inequality is needed, so poses in which the two shadows touch need no special treatment.