5 The Global Theorem

Lemma 26

✓

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

\[ \langle S ,w \rangle \leq \max _{i} \langle V_i ,w\rangle \]

Proof ▶

This is a mild generalization of [ SY25 ] , Lemma 18.

Since \(S \in \mathsf{Co}(V)\), we have

\[ S = \sum _{j=1}^m \lambda _j V_j \]

for some \(\lambda _1,\ldots ,\lambda _m \in [0,1]\) with

\[ 1 = \sum _{j=1}^m \lambda _j \]

Therefore

\[ \langle S ,w \rangle = \left\langle \sum _{j=1}^m \lambda _j V_j ,w \right\rangle = \sum _{j=1}^m \lambda _j \left\langle V_j ,w \right\rangle \le \sum _{j=1}^m \lambda _j \max _{i} \langle V_i ,w\rangle \]

\[ = \max _{i} \langle V_i ,w\rangle \sum _{j=1}^m \lambda _j = \max _{i} \langle V_i ,w\rangle \]

as required.

Lemma 27

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

\[ \left|\frac{\mathrm{d}^2 f}{\mathrm{d}x_i \mathrm{d}x_j}(x_1,x_2,x_3)\right|\leq 1. \]

Proof ▶

See [ SY25 ] , Lemma 19.

Lemma 28

✓

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 \( \left|\partial _{x_i}\partial _{x_j}f(v)\right| \leq 1 \) for all \(i,j \in \{ 1,\dots ,n\} \) and all \(v \in \operatorname{\mathbb {R}}^n\), then

\[ |f(x_1,\dots ,x_n)-f(y_1,\dots ,y_n)|\leq \varepsilon \sum _{i=1}^n |\partial _{x_i} f(x_1,\dots ,x_n)| + \frac{n^2}{2}\varepsilon ^2. \]

Proof ▶

See [ SY25 ] , Lemma 20.

Lemma 29

The partial derivatives of all relevant rotations, projections, and inner products used in the Global Theorem are as expected. Specifically:

\[ f^\alpha (\theta ,\varphi ,\alpha ) = \langle R'(\alpha ) M(\theta , \varphi ) S, w \rangle \]
\[ f^\theta (\theta ,\varphi ,\alpha ) = \langle R(\alpha ) M^\theta (\theta , \varphi ) S, w \rangle \]
\[ f^\varphi (\theta ,\varphi ,\alpha ) = \langle R(\alpha ) M^\varphi (\theta , \varphi ) S, w \rangle \]
\[ g^\theta (\theta ,\varphi ) = \langle M^\theta (\theta , \varphi ) P, w \rangle \]
\[ g^\varphi (\theta ,\varphi ) = \langle M^\varphi (\theta , \varphi ) P, w \rangle \]

where \(f(\theta ,\varphi ,\alpha ) = \langle R(\alpha ) M(\theta ,\varphi ) S / \| S\| , w\rangle \) and \(g(\theta ,\varphi ) = \langle M(\theta ,\varphi ) P / \| P\| , w\rangle \).

Proof ▶

By basic properties of derivatives.

Theorem 30 Global Theorem

✓

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

\begin{align*} G& :=\langle R(\overline{\alpha }) \overline{M_1}S,w \rangle - \varepsilon \cdot \big(|\langle R’(\overline{\alpha }) \overline{M_1}S,w \rangle |+|\langle R(\overline{\alpha }) \overline{M_1}^\theta S,w \rangle |+|\langle R(\overline{\alpha }) \overline{M_1}^\varphi S,w \rangle |\big)- 9\varepsilon ^2/2,\\ H_P & :=\langle \overline{M_2}P,w \rangle + \varepsilon \cdot \big(|\langle \overline{M_2}^\theta P,w \rangle |+|\langle \overline{M_2}^\varphi P,w \rangle |\big) + 2\varepsilon ^2, \quad \text{ for } P \in \operatorname{\mathbf{P}}. \end{align*}

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

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

Proof ▶

See [ SY25 ] , Section 4.2.