6 The Local Theorem

Lemma 29

✓

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

Proof ▶

See [ SY25 ] , Lemma 21.

Definition 30

✓

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

\[ \mathrm{span}^+(v_1,\dots ,v_n) = \Big\{ w \in \operatorname{\mathbb {R}}^n \colon \exists \lambda _1,\dots ,\lambda _n {\gt} 0 \text{ s.t. } w = \sum _{i=1}^n \lambda _i v_i \Big\} , \]

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

Lemma 31

✓

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:

\begin{align*} \langle V_1, Y \rangle {\gt} \langle V_1, Z \rangle ,\\ \langle V_2, Y \rangle {\gt} \langle V_2, Z \rangle ,\\ \langle V_3, Y \rangle {\gt} \langle V_3, Z \rangle . \end{align*}

Proof ▶

See [ SY25 ] , Lemma 23.

Lemma 32

✓

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

\[ |\langle AP_1,BP_2\rangle -\langle \overline{A}P_1,\overline{B}P_2\rangle |\leq \| P_1\| \cdot \| P_2\| \cdot \Big( \| A-\overline{A}\| \cdot \| \overline{B}\| + \| \overline{A}\| \cdot \| B-\overline{B}\| +\| A-\overline{A}\| \cdot \| B-\overline{B}\| \Big). \]

Proof ▶

See [ SY25 ] , Lemma 24.

Lemma 33

✓

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

\[ |\langle AP_1,AP_2\rangle -\langle BP_1,BP_2\rangle |\leq \| P_1\| \cdot \| P_2\| \cdot \| A-B\| \cdot \bigg(\| A\| +\| B\| + \| A-B\| \bigg). \]

Proof ▶

See [ SY25 ] , Lemma 25.

Lemma 34

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 $ABC$.

Proof ▶

See [ SY25 ] , Lemma 26.

Definition 35

✓

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:

\begin{align*} \langle R(\pi /2) M P_1,M P_{2}\rangle {\gt} 2 \varepsilon (\sqrt{2} + \varepsilon ),\\ \langle R(\pi /2) M P_2,M P_{3}\rangle {\gt} 2 \varepsilon (\sqrt{2} + \varepsilon ),\\ \langle R(\pi /2) M P_3,M P_{1}\rangle {\gt} 2 \varepsilon (\sqrt{2} + \varepsilon ). \end{align*}

Lemma 36

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

\[ X(\theta , \varphi ) \in \operatorname{\mathrm{span}^+}(P_1, P_2, P_3). \]

Proof ▶

See [ SY25 ] , Lemma 28.

Lemma 37

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

\[ T :=\left(R(\overline{\alpha }) M(\overline{\theta }_1, \overline{\varphi }_1) P + M(\overline{\theta }_2, \overline{\varphi }_2) Q\right)/2 \in \operatorname{\mathbb {R}}^2, \]

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 $R(\alpha )M(\theta _1, \varphi _1) P, M(\theta _2, \varphi _2) Q \in \operatorname{\mathrm{Disc}}_{\delta + \sqrt{5} \varepsilon }(T)$.

Proof ▶

See [ SY25 ] , Lemma 30.

Definition 38

✓

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\| $.

Lemma 39

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

\[ \frac{\langle Q, Q - P_j \rangle }{\| Q\| \| Q - P_j\| } \geq \delta /r, \]

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 $\overline{Q}$.

Proof ▶

See [ SY25 ] , Lemma 32.

Lemma 40

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:

\[ \frac{\langle {M} P,{M} (P-Q)\rangle }{\| {M} P\| \cdot \| {M}(P-Q)\| } \geq \frac{\langle \overline{M} P,\overline{M} (P-Q)\rangle - 2 \varepsilon \| P-Q\| \cdot (\sqrt{2}+\varepsilon )}{ \big(\| \overline{M} P\| +\sqrt{2} \varepsilon \big) \cdot \big(\| \overline{M}(P-Q)\| +2 \sqrt{2} \varepsilon \big)}. \]

Proof ▶

See [ SY25 ] , Lemma 33.

Theorem 41 Local Theorem

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

\[ (-1)^{\sigma _P} \langle \overline{X_1},P_i\rangle {\gt}\sqrt{2}\varepsilon \quad \text{and} \quad (-1)^{\sigma _Q} \langle \overline{X_2}, Q_i\rangle {\gt}\sqrt{2}\varepsilon , \tag {A$_\varepsilon $} \]

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

\[ \frac{\langle \overline{M_2}Q_i,\overline{M_2}(Q_i-Q_j)\rangle - 2 \varepsilon \| Q_i-Q_j\| \cdot (\sqrt{2}+\varepsilon )}{ \big(\| \overline{M_2}Q_i\| +\sqrt{2} \varepsilon \big) \cdot \big(\| \overline{M_2}(Q_i-Q_j)\| +2 \sqrt{2} \varepsilon \big)} {\gt} \frac{\sqrt{5} \varepsilon + \delta }{r}, \tag {B$_\varepsilon $} \]

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

\[ \delta \geq \max _{i=1,2,3}\left\| R(\overline{\alpha }) \overline{M_1}P_i - \overline{M_2}Q_i\right\| /2. \]

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

\[ (\theta _1, \varphi _1, \theta _2, \varphi _2, \alpha ) \in [\overline{\theta }_1\pm \varepsilon ,\overline{\varphi }_1\pm \varepsilon ,\overline{\theta }_2\pm \varepsilon ,\overline{\varphi }_2\pm \varepsilon ,\overline{\alpha }\pm \varepsilon ] :=U \subseteq \operatorname{\mathbb {R}}^5. \]

Proof ▶

See [ SY25 ] , Theorem 36.