Dependencies

\(\| C_1 \| = 1\), \({98 \over 100} {\lt} \| C_2 \| {\lt} {99 \over 100}\), and \({98 \over 100} {\lt} \| C_3 \| {\lt} {99 \over 100}\).

If the noperthedron is Rupert, then there exists a solution with

Let \(\alpha ,\theta ,\varphi \in [-4,4]\). Then it holds that

Moreover,

Let \(\theta , \varphi \in \operatorname{\mathbb {Q}}\cap [-4,4]\) and \(M_{\operatorname{\mathbb {Q}}} :=M_{\operatorname{\mathbb {Q}}}(\theta , \varphi )\). Three points \(\widetilde{P}_1, \widetilde{P}_2, \widetilde{P}_3 \in \operatorname{\mathbb {Q}}^3\) with \(\| \widetilde{P}_1\| , \| \widetilde{P}_2\| , \| \widetilde{P}_3\| \leq 1+\kappa \) are called \(\varepsilon \)-\(\kappa \)-spanning for \((\theta , \varphi )\) if it holds that:

Let \(\theta , \varphi \in \mathbb {R}\), \(\varepsilon {\gt} 0\), and set \(M := M(\theta , \varphi )\). Three points \(P_1, P_2, P_3 \in \mathbb {R}^3\) with \(\| P_1\| , \| P_2\| , \| P_3\| \leq 1\) are called \(\varepsilon \)-spanning for \((\theta , \varphi )\) if it holds that:

Let \(\operatorname{\mathcal{P}}\subset \operatorname{\mathbb {R}}^2\) be a convex polygon and \(Q \in \operatorname{\mathcal{P}}\) one of its vertices. Assume that for some \(\overline{Q} \in \operatorname{\mathbb {R}}^2\) it holds that \(Q \in \operatorname{\mathrm{Disc}}_{\delta }(\overline{Q})\), i.e. \(\| Q - \overline{Q}\| {\lt} \delta \). Define \(\operatorname{\mathrm{Sect}}_\delta (\overline{Q}) :=\operatorname{\mathrm{Disc}}_{\delta }(\overline{Q}) \cap \operatorname{\mathcal{P}}^\circ \) as the intersection between \(\operatorname{\mathrm{Disc}}_{\delta }(\overline{Q})\) and the interior of the convex hull of \(\operatorname{\mathcal{P}}\).

Moreover, \(Q \in \operatorname{\mathcal{P}}\) is called \(\delta \)-locally maximally distant with respect to \(\overline{Q}\) (\(\delta \)-LMD\((\overline{Q})\)) if for all \(A \in \operatorname{\mathrm{Sect}}_\delta (\overline{Q})\) it holds that \(\| Q\| {\gt} \| A\| \).

The Noperthedron is polyhedron given by the vertex set that is the pointsymmetrization of

A set \(S \subseteq \operatorname{\mathbb {R}}^3\) is point-symmetric if \(x \in S\) implies \(-x \in S\).

The pointsymmetrization of a collection of vertices \(v_1, \ldots , v_n \in \operatorname{\mathbb {R}}^3\) is \(v_1, \ldots , v_n, -v_1, \ldots , -v_n\).

Given \(v_1, \dots , v_n \in \operatorname{\mathbb {R}}^n\) write \(\mathrm{span}^+(v_1,\dots ,v_n)\) for the set (simplicial cone) in \(\operatorname{\mathbb {R}}^n\) defined by

which is the natural restriction of \(\mathrm{span}(v_1,\dots ,v_n)\) to positive weights.

We define the two functions \(\sin _{\mathbb {Q}}, \cos _{\mathbb {Q}}: \operatorname{\mathbb {R}}\to \operatorname{\mathbb {R}}\) by:

Further, by replacing \(\sin ,\cos \) with \(\sin _{\mathbb {Q}},\cos _{\mathbb {Q}}\) we define the functions

In the setting of lemma 53 let additionally \(\overline{\theta }, \overline{\varphi }\in \operatorname{\mathbb {R}}\cap [-4,4]\) and set \(\overline{M} = M(\overline{\theta }, \overline{\varphi })\), \(\overline{M}_{\operatorname{\mathbb {Q}}} = M_{\operatorname{\mathbb {Q}}}(\overline{\theta }, \overline{\varphi })\). Then

For \(1 \leq i \leq n\) let \((A_i,B_i)\) be pairs of real matrices, such that for each \(i\) the dimensions of \(A_i\) and \(B_i\) are equal. Assume moreover that the products \(A_1\cdots A_n\) and \(B_1 \cdots B_n\) are well defined. Finally, assume that \(\| A_i-B_i\| \leq \kappa \) and let \(\delta _i\geq \max (\| A_i\| ,\| B_i\| ,1)\). Then it holds that \(\| A_1\cdots A_n-B_1\cdots B_n\| \leq n\kappa \cdot \delta _1\cdots \delta _n\).

Let \(A = (a_{i,j})_{1 \leq i \leq m,\ 1 \leq j \leq n} \in \operatorname{\mathbb {R}}^{m \times n}\) and \(\delta {\gt}0\). Assume that \(|a_{i,j}| \leq \delta \). Then it holds that \(\| A\| \leq \delta \sqrt{mn}.\)

For \(A,B\in \operatorname{\mathbb {R}}^{n\times n}\) and \(P_1,P_2\in \operatorname{\mathbb {R}}^n\) one has

Let \(\alpha , \theta , \varphi \in [-4,4]\), \(P\in \operatorname{\mathbb {R}}^3\) with \(\| P\| \leq 1\) and let \(\widetilde{P}\) be a \(\kappa \)-rational approximation of \(P\). Set \(M = M(\theta , \varphi )\) and \(M_{\operatorname{\mathbb {Q}}} = M_{\operatorname{\mathbb {Q}}}(\theta , \varphi )\), \(M^\theta = M^\theta (\theta , \varphi )\), \(M^\theta _{\operatorname{\mathbb {Q}}} = M^\theta _{\operatorname{\mathbb {Q}}}(\theta , \varphi )\), \(M^\varphi = M^\varphi (\theta , \varphi )\), \(M^\varphi _{\operatorname{\mathbb {Q}}} = M^\varphi _{\operatorname{\mathbb {Q}}}(\theta , \varphi )\) as well as \(R = R(\alpha )\), \(R_{\operatorname{\mathbb {Q}}} = R_{\operatorname{\mathbb {Q}}}(\alpha )\), \(R' = R'(\alpha )\), \(R'_{\operatorname{\mathbb {Q}}} = R'_{\operatorname{\mathbb {Q}}}(\alpha )\). Finally let \(w \in \operatorname{\mathbb {R}}^2\) with \(\| w\| = 1\). Then:

Let \(P,Q \in \operatorname{\mathbb {R}}^3\) with \(\| P\| ,\| Q\| \leq 1\) and \(\widetilde{P},\widetilde{Q}\) some respective \(\kappa \)-rational approximations. Moreover, let \(\alpha , \theta , \varphi \in \operatorname{\mathbb {R}}\in [-4,4]\) and set \(X = X(\theta , \varphi )\), \(X_{\operatorname{\mathbb {Q}}} = X_{\operatorname{\mathbb {Q}}}(\theta , \varphi )\) as well as \(M = M(\theta , \varphi )\), \(M_{\operatorname{\mathbb {Q}}} = M_{\operatorname{\mathbb {Q}}}(\theta , \varphi )\). Then

In the setting of lemma 53, let \(\sqrt[+]{x}\) be an upper-\(\operatorname{\mathbb {Q}}\)-square-root function and set \(\| x\| _{+} :=\sqrt[+]{\| x\| ^2}\). Set

as well as

Then it holds that \(A \geq A_{\operatorname{\mathbb {Q}}}\).

Let \(\varepsilon {\gt}0\) and \(\theta ,\overline{\theta }, \varphi , \overline{\varphi }\in \operatorname{\mathbb {R}}\) with \(|\theta - \overline{\theta }|, |\varphi - \overline{\varphi }| \leq \varepsilon \). Define \(M = M(\theta , \varphi )\) and \(\overline{M} = M(\overline{\theta }, \overline{\varphi })\) and let \(P, Q \in \operatorname{\mathbb {R}}^3\) with \(\| P\| , \| Q\| \leq 1\). Then:

Let \(A(x,y)\) be an \(m\times n\) matrix with \(1 \leq m,n\leq 3\) such that every entry is in \([-1,1]\).of the form \(a_1(x)\cdot a_2(y)\) where \(a_i(z) \in [-1,1]\). Define \(A_{\mathbb {Q}}(x,y)\) by replacing \(\sin \) with \(\sin _{\mathbb {Q}}\) and \(\cos \) with \(\cos _{\mathbb {Q}}\). Then for every \(x,y\in [-4,4]\) it holds that \(\| A(x,y)-A_{\mathbb {Q}}(x,y)\| \leq \kappa \).

Let \(P_1, P_2, P_3 \in \operatorname{\mathbb {R}}^3\) with \(\| P_i\| \leq 1\) and \(\widetilde{P}_1, \widetilde{P}_2, \widetilde{P}_3 \in \operatorname{\mathbb {Q}}^3\) be their \(\kappa \)-rational approximations. Assume that \(\widetilde{P}_1, \widetilde{P}_2, \widetilde{P}_3\) are \(\varepsilon \)-\(\kappa \)-spanning for some \(\theta , \varphi \in \operatorname{\mathbb {Q}}\cap [-4,4]\), then \(P_1, P_2, P_3\) are \(\varepsilon \)-spanning for \(\theta , \varphi \).

Let \(P_1, P_2, P_3 \in \operatorname{\mathbb {R}}^3\) with \(\| P_1\| ,\| P_2\| ,\| P_3\| \leq 1\) be \(\varepsilon \)-spanning for \((\overline{\theta }, \overline{\varphi })\) and let \(\theta , \varphi \in \operatorname{\mathbb {R}}\) such that \(|\theta - \overline{\theta }|, |\varphi - \overline{\varphi }| \leq \varepsilon \). Assume that \(\langle X(\theta , \varphi ), P_i \rangle {\gt} 0\) for \(i=1,2,3\). Then

Suppose \(V = V_1, \ldots , V_m \subseteq \operatorname{\mathbb {R}}^n\) be a finite sequence of points. Suppose \(\mathsf{Co}(V)\) is its convex hull. Let \(S \in \mathsf{Co}(V)\) and \(w \in \operatorname{\mathbb {R}}^n\) be given. then

Let \(P, Q \in \operatorname{\mathbb {R}}^3\) with \(\| P\| , \| Q\| \leq 1\). Let \(\varepsilon {\gt}0\) and \(\overline{\theta }_1,\overline{\varphi }_1,\overline{\theta }_2,\overline{\varphi }_2,\overline{\alpha }\in \operatorname{\mathbb {R}}\), then set

and \(\delta \geq \| T - M(\overline{\theta }_2, \overline{\varphi }_2) Q\| \). Finally, let \(\theta _1, \varphi _1, \theta _2, \varphi _2, \alpha \in \operatorname{\mathbb {R}}\) with \(|\overline{\theta }_1-\theta _1|, |\overline{\varphi }_1 - \varphi _1|, |\overline{\theta }_2-\theta _2|, |\overline{\varphi }_2-\varphi _2|, |\overline{\alpha }- \alpha | \leq \varepsilon \). Then \(R(\alpha )M(\theta _1, \varphi _1) P, M(\theta _2, \varphi _2) Q \in \operatorname{\mathrm{Disc}}_{\delta + \sqrt{5} \varepsilon }(T)\).

For all \(a,b \in \operatorname{\mathbb {R}}\) with \(|a|,|b|\leq 2\) the following inequality holds:

with equality only for \(a=0\) or \(b=0\).

For every \(x\in [-4,4]\) it holds that

Let \(V_1,V_2,V_3,Y,Z \in \operatorname{\mathbb {R}}^3\) with \(\| Y \| =\| Z \| \) and \(Y,Z \in \mathrm{span}^+(V_1,V_2,V_3)\). Then at least one of the following inequalities does not hold:

Let \(S \in \operatorname{\mathbb {R}}^3\) and \(w \in \operatorname{\mathbb {R}}^2\) be unit vectors and set \(f(x_1,x_2,x_3) = \langle R(x_3) M(x_1,x_2)S,w \rangle \). Then for all \(x_1,x_2,x_3 \in \operatorname{\mathbb {R}}\) and any \(i,j \in \{ 1,2,3\} \) it holds that

Let \(\operatorname{\mathcal{P}}\) be a convex polygon and \(Q \in \operatorname{\mathcal{P}}\) be one of its vertices. Let \(\overline{Q} \in \operatorname{\mathbb {R}}^2\) with \(\| Q - \overline{Q}\| {\lt} \delta \) for some \(\delta {\gt}0\). Assume that for some \(r {\gt} 0 \) such that \(\| Q\| {\gt} r\) it holds that

for all other vertices \(P_j \in \operatorname{\mathcal{P}}\setminus Q\). Then \(Q \in \operatorname{\mathcal{P}}\) is \(\delta \)-locally maximally distant with respect to \(\overline{Q}\).

Let \(P \in \operatorname{\mathbb {R}}^3\) with \(\| P\| \leq 1\). Further, let \(\varepsilon , r{\gt}0\) and \(\overline{\theta },\overline{\varphi }, \theta , \varphi \in \operatorname{\mathbb {R}}\) such that \(|\overline{\theta }-\theta |, |\overline{\varphi }- \varphi | \leq \varepsilon \). If \( \| M(\overline{\theta },\overline{\varphi }) P \| {\gt} r + \sqrt{2}\varepsilon \) then \( \| M(\theta ,\varphi ) P \| {\gt} r. \)

Let \(f:\operatorname{\mathbb {R}}^n\to \operatorname{\mathbb {R}}\) be a \(C^2\)-function and \(x_1,\dots ,x_n,y_1,\dots ,y_n \in \operatorname{\mathbb {R}}\) such that \(|x_1-y_1|,\dots ,|x_n-y_n|\leq \varepsilon \). If \( \left|\partial _{x_i}\partial _{x_j}f(x_1,\dots ,x_n)\right| \leq 1 \) for all \(i,j \in \{ 1,\dots ,n\} \) then

Let \(A,B,C\in \mathbb {R}^2\) be such that \( \langle R({\pi /2}) A,B\rangle , \langle R({\pi /2}) B,C\rangle , \langle R({\pi /2}) C,A\rangle {\gt}0\). Then the origin lies strictly in the triangle \(ABC\).

For any \(P \in \mathbb {R}^3\) one has \(\big\| M(\theta , \varphi ) P\big\| ^2=\| P\| ^2-\langle X({\theta ,\varphi }),P\rangle ^2\).

The radius of the Noperthedron is one.

Let \(\varepsilon {\gt}0\), \(|\alpha -\overline{\alpha }|\leq \varepsilon \) and \(a \in \{ x,y,z\} \) then \(\| R_a(\alpha )-R_a({\overline{\alpha }})\| =\| R(\alpha )-R(\overline{\alpha })\| {\lt} \varepsilon \).

For any \(\alpha , \theta ,\varphi \in \operatorname{\mathbb {R}}\) and \(a \in \{ x,y,z\} \) one has \(\| R(\alpha )\| = \| R_a(\alpha )\| =\| M(\theta , \varphi )\| = 1\).

For any \(\alpha ,\beta \in \mathbb {R}\) one has

with equality only for \(\alpha = \beta = 0\).

For \(A,\overline{A},B,\overline{B}\in \operatorname{\mathbb {R}}^{n\times n}\) and \(P_1,P_2\in \operatorname{\mathbb {R}}^n\) it holds that

Let \(\varepsilon {\gt}0\) and \(|\theta -\overline{\theta }|,|\varphi -\overline{\varphi }| \leq \varepsilon \) then \(\| M(\theta , \varphi )-M(\overline{\theta },\overline{\varphi })\| , \| X({\theta , \varphi })-X(\overline{\theta },\overline{\varphi })\| {\lt} \sqrt{2}\varepsilon .\)

Let \(\varepsilon {\gt}0\) and \(|\theta -\overline{\theta }|,|\varphi -\overline{\varphi }|,|\alpha -\overline{\alpha }|\leq \varepsilon \) then \(\| R(\alpha ) M(\theta , \varphi )-R(\overline{\alpha })M(\overline{\theta },\overline{\varphi })\| {\lt} \sqrt{5} \varepsilon .\)

Let \(\operatorname{\mathbf{P}}= \operatorname{\mathbf{NOP}}\), then for all \(\theta , \varphi , \alpha \in \operatorname{\mathbb {R}}\), the following three identities hold (as sets):

Let \(P \in \operatorname{\mathbb {R}}^3\) with \(\| P\| \leq 1\). Further, let \(\varepsilon {\gt}0\) and \(\overline{\theta },\overline{\varphi }, \theta , \varphi \in \operatorname{\mathbb {R}}\) such that \(|\overline{\theta }-\theta |, |\overline{\varphi }- \varphi | \leq \varepsilon \). If \( \langle X(\overline{\theta },\overline{\varphi }),P \rangle {\gt}\sqrt{2}\varepsilon \) then \( \langle X(\theta , \varphi ),P \rangle {\gt}0. \)

The norm of any vertex in the prepointsymmetrized version of the Noperthedron is no more than 1.

The noperthedron is point-symmetric.

The pointsymmetrization of any set is point-symmetric.

There exists a valid solution table with some row that covers

Let \(\operatorname{\mathbf{P}}\) be a pointsymmetric convex polyhedron with radius \(\rho =1\) and let \(S \in \operatorname{\mathbf{P}}\). Further let \(\overline{\theta }_1,\overline{\varphi }_1,\overline{\theta }_2,\overline{\varphi }_2,\overline{\alpha }\in \operatorname{\mathbb {R}}\) and let \(w\in \operatorname{\mathbb {R}}^2\) be a unit vector. Denote \(\overline{M_1}:=M(\overline{\theta }_1, \overline{\varphi }_1)\), \( \overline{M_2}:=M(\overline{\theta }_2, \overline{\varphi }_2)\) as well as \(\overline{M_1}^{\theta } :=M^\theta (\overline{\theta }_1, \overline{\varphi }_1)\), \(\overline{M_1}^{\varphi } :=M^\varphi (\overline{\theta }_1, \overline{\varphi }_1)\) and analogously for \(\overline{M_2}^{\theta }, \overline{M_2}^{\varphi }\). Finally set

If \(G{\gt}\max _{P\in \operatorname{\mathbf{P}}} H_P\) then there does not exist a solution to Rupert’s condition with

Let \(\operatorname{\mathbf{P}}\) be a pointsymmetric convex polyhedron with radius \(\rho =1\) and \(\widetilde{\operatorname{\mathbf{P}}}\) a \(\kappa \)-rational approximation. Let \(\widetilde{S} \in \widetilde{\operatorname{\mathbf{P}}}\). Further let \(\varepsilon {\gt}0\) and \(\overline{\theta }_1,\overline{\varphi }_1,\overline{\theta }_2,\overline{\varphi }_2,\overline{\alpha }\in \operatorname{\mathbb {Q}}\cap [-4,4]\). Let \(w\in \operatorname{\mathbb {Q}}^2\) be a unit vector. Denote \(\overline{M_1}:=M_{\operatorname{\mathbb {Q}}}(\overline{\theta }_1, \overline{\varphi }_1)\), \( \overline{M_2}:=M_{\operatorname{\mathbb {Q}}}(\overline{\theta }_2, \overline{\varphi }_2)\) as well as \(\overline{M_1}^{\theta } :=M_{\operatorname{\mathbb {Q}}}^\theta (\overline{\theta }_1, \overline{\varphi }_1)\), \(\overline{M_1}^{\varphi } :=M_{\operatorname{\mathbb {Q}}}^\varphi (\overline{\theta }_1, \overline{\varphi }_1)\) and analogously for \(\overline{M_2}^{\theta }, \overline{M_2}^{\varphi }\). Finally set

If \(G^{\operatorname{\mathbb {Q}}}{\gt}\max _{P\in \widetilde{\operatorname{\mathbf{P}}}} H^{\operatorname{\mathbb {Q}}}_P\) then there does not exist a solution to Rupert’s condition to \(\operatorname{\mathbf{P}}\) with

Let \(\operatorname{\mathbf{P}}\) be a polyhedron with radius \(\rho =1\) and \(P_1, P_2, P_3, Q_1, Q_2, Q_3 \in \operatorname{\mathbf{P}}\) be not necessarily distinct. Assume that \(P_1, P_2, P_3\) and \(Q_1, Q_2, Q_3\) are congruent.

Let \(\varepsilon {\gt}0\) and \(\overline{\theta }_1,\overline{\varphi }_1,\overline{\theta }_2,\overline{\varphi }_2,\overline{\alpha }\in \operatorname{\mathbb {R}}\), then set \(\overline{X_1}:=X(\overline{\theta }_1,\overline{\varphi }_1), \overline{X_2}:=X(\overline{\theta }_2,\overline{\varphi }_2)\) as well as \(\overline{M_1}:=M(\overline{\theta }_1,\overline{\varphi }_1), \overline{M_2}:=M(\overline{\theta }_2,\overline{\varphi }_2)\). Assume that there exist \(\sigma _P, \sigma _Q \in \{ 0,1\} \) such that

for all \(i=1,2,3\). Moreover, assume that \(P_1,P_2,P_3\) are \(\varepsilon \)-spanning for \((\overline{\theta }_1,\overline{\varphi }_1)\) and that \(Q_1,Q_2,Q_3\) are \(\varepsilon \)-spanning for \((\overline{\theta }_2,\overline{\varphi }_2)\). Finally, assume that for all \(i = 1,2,3\) and any \(Q_j \in \operatorname{\mathbf{P}}\setminus Q_i\) it holds that

for some \(r {\gt}0\) such that \(\min _{i=1,2,3}\| \overline{M_2}Q_i \| {\gt} r + \sqrt{2} \varepsilon \) and for some \(\delta \in \operatorname{\mathbb {R}}\) with

Then there exists no solution to Rupert’s problem \(R(\alpha ) M(\theta _1,\varphi _1)\operatorname{\mathbf{P}}\subset M(\theta _2,\varphi _2)\operatorname{\mathbf{P}}^\circ \) with

Let \(\operatorname{\mathbf{P}}\) be a polyhedron with radius \(\rho =1\) and \(\widetilde{P}_i\) be a \(\kappa \)-rational approximation of \(P_i \in \operatorname{\mathbf{P}}\). Set \(\widetilde{\operatorname{\mathbf{P}}} = \{ \widetilde{P}_i \text{ for } P_i \in \operatorname{\mathbf{P}}\} \). Let \(P_1, P_2, P_3, Q_1, Q_2, Q_3 \in \operatorname{\mathbf{P}}\) be not necessarily distinct and assume that \(P_1, P_2, P_3\) and \(Q_1, Q_2, Q_3\) are congruent. Let \(\varepsilon {\gt}0\) and \(\overline{\theta }_1,\overline{\varphi }_1,\overline{\theta }_2,\overline{\varphi }_2,\overline{\alpha }\in \operatorname{\mathbb {Q}}\cap [-4,4]\). Set \(\overline{X_1}:=X_{\operatorname{\mathbb {Q}}}(\overline{\theta }_1,\overline{\varphi }_1), \overline{X_2}:=X_{\operatorname{\mathbb {Q}}}(\overline{\theta }_2,\overline{\varphi }_2)\) as well as \(\overline{M_1}:=M_{\operatorname{\mathbb {Q}}}(\overline{\theta }_1,\overline{\varphi }_1), \overline{M_2}:=M_{\operatorname{\mathbb {Q}}}(\overline{\theta }_2,\overline{\varphi }_2)\). Assume that there exist \(\sigma _P, \sigma _Q \in \{ 0,1\} \) such that

for all \(i=1,2,3\). Moreover, assume that \(\widetilde{P}_1,\widetilde{P}_2,\widetilde{P}_3\) are \(\varepsilon \)-\(\kappa \)-spanning for \((\overline{\theta }_1,\overline{\varphi }_1)\) and that \(\widetilde{Q}_1,\widetilde{Q}_2,\widetilde{Q}_3\) are \(\varepsilon \)-\(\kappa \)-spanning for \((\overline{\theta }_2,\overline{\varphi }_2)\). Let \(\sqrt[+]{x}\) and \(\sqrt[-]{x}\) be upper- and lower-\(\operatorname{\mathbb {Q}}\)-square-root functions, then set \(\| Z\| _{+} :=\sqrt[+]{\| Z\| ^2}\) and \(\| Z\| _{-} :=\sqrt[-]{\| Z\| ^2}\) for \(Z \in \operatorname{\mathbb {Q}}^n\). Finally, assume that for all \(i = 1,2,3\) and any \(\widetilde{Q}_j \in \widetilde{\operatorname{\mathbf{P}}} \setminus \widetilde{Q}_i\) it holds that

for some \(r {\gt}0\) such that \(\min _{i=1,2,3}\| \overline{M_2}\widetilde{Q}_i \| _{-} {\gt} r + \sqrt{2} \varepsilon + 3\kappa \) and for some \(\delta \in \operatorname{\mathbb {R}}\) with

Then there exists no solution to Rupert’s problem \(R(\alpha ) M(\theta _1,\varphi _1)\operatorname{\mathbf{P}}\subset M(\theta _2,\varphi _2)\operatorname{\mathbf{P}}^\circ \) with

There is no pose that makes the noperthedron have the Rupert property.

There is no purely rotational pose that makes the noperthedron have the Rupert property.

There does not in fact exist a noperthedron Rupert solution with

There is no view pose that makes the noperthedron have the Rupert property.

The noperthedron is not a Rupert polyhedron.

The noperthedron is not a Rupert set.

Pointsymmetrization preserves radius.

Suppose \(S\) is a finite set of points in \(\operatorname{\mathbb {R}}^n\). The radius of the polyhedron \(S\) is \(r\) iff

• there is a vector \(v \in S\) with \(\| v\| = r\)

• all vectors \(v \in S\) have \(\| v\| \le r\)

For any valid row in a valid solution table, there can be no Rupert solution in the pose interval of that row.

The following are equivalent:

• The convex polyhedron with vertex set \(v\) is Rupert.

• The convex closure of \(v\) is a Rupert set.

If a set is point symmetric and convex, then it being Rupert implies it being purely rotationally Rupert.

Given a pose with zero offset, there exists a view pose that is equivalent to it.