5 The Global Theorem

Lemma 18

✓

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 19

✓

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 \(i,j,k \in \{ 1,2,3\} \) it holds that

\[ \left|\frac{\mathrm{d}^2 f}{\mathrm{d}x_i \mathrm{d}x_j}(x_1,x_2,x_3)\right|\leq 1 \qquad \text{and}\qquad \left|\frac{\mathrm{d}^3 f}{\mathrm{d}x_i \mathrm{d}x_j \mathrm{d}x_k}(x_1,x_2,x_3)\right|\leq 1. \]

Proof ▶

The second-partial bound is [ SY25 ] , Lemma 19. The third-partial bound follows by the same argument: every third partial of \(f\) is \(\langle A S, w\rangle \) for \(A\) a composition of \(\pm R\) or \(\pm R'\) with a matrix from the \(M\)-derivative family, all of operator norm at most one. (Note \(M^{\theta \theta \theta } = -M^\theta \) and \(M^{\varphi \varphi \varphi } = -M^\varphi \), so only the two mixed third derivatives \(M^{\theta \theta \varphi }\), \(M^{\theta \varphi \varphi }\) are genuinely new matrices.)

Lemma 20

✓

Let \(f:\operatorname{\mathbb {R}}^n\to \operatorname{\mathbb {R}}\) be a \(C^3\)-function, let \(\varepsilon _1,\dots ,\varepsilon _n \geq 0\), and let \(x_1,\dots ,x_n,y_1,\dots ,y_n \in \operatorname{\mathbb {R}}\) be such that \(|x_i-y_i|\leq \varepsilon _i\) for all \(i\). If \( \left|\partial _{x_i}\partial _{x_j}\partial _{x_k}f(v)\right| \leq 1 \) for all \(i,j,k \in \{ 1,\dots ,n\} \) and all \(v \in \operatorname{\mathbb {R}}^n\), then

\[ |f(x)-f(y)|\leq \sum _{i=1}^n \varepsilon _i |\partial _{x_i} f(x)| + \frac{1}{2} \sum _{i=1}^n \sum _{j=1}^n \varepsilon _i \varepsilon _j |\partial _{x_i}\partial _{x_j} f(x)| + \frac{1}{6}\Big(\sum _{i=1}^n \varepsilon _i\Big)^3. \]

(At \(\varepsilon _1 = \dots = \varepsilon _n = \varepsilon \) this recovers the isotropic remainder \(\frac{n^3}{6}\varepsilon ^3\).)

Proof ▶

This strengthens [ SY25 ] , Lemma 20, by one Taylor order and per-axis radii: expand \(g(t) = f((1-t)x + ty)\) to second order with Lagrange remainder, bound \(|g'(0)|\) by the first sum, \(|g''(0)|/2\) by the second (the second partials are evaluated exactly at \(x\), not bounded), and \(|g'''(c)| \le (\sum _i \varepsilon _i)^3\) using the third-partial bound.

Lemma 21

✓

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 22 Global Theorem

✓

Let \(\operatorname{\mathbf{P}}\) be a pointsymmetric convex polyhedron with radius \(\rho =1\) and let \(S \in \operatorname{\mathbf{P}}\). Further let \(\overline{\theta }_1,\overline{\varphi }_1,\overline{\theta }_2,\overline{\varphi }_2,\overline{\alpha }\in \operatorname{\mathbb {R}}\), let \(\varepsilon _\alpha , \varepsilon _{\theta _1}, \varepsilon _{\varphi _1}, \varepsilon _{\theta _2}, \varepsilon _{\varphi _2} \geq 0\) be per-axis radii, 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 _\alpha |\langle R’(\overline{\alpha }) \overline{M_1}S,w \rangle | - \varepsilon _{\theta _1}|\langle R(\overline{\alpha }) \overline{M_1}^\theta S,w \rangle | - \varepsilon _{\varphi _1}|\langle R(\overline{\alpha }) \overline{M_1}^\varphi S,w \rangle |\\ & - \frac{1}{2}\big(\varepsilon _\alpha ^2|\langle R(\overline{\alpha }) \overline{M_1}S,w \rangle | + 2\varepsilon _\alpha \varepsilon _{\theta _1}|\langle R’(\overline{\alpha }) \overline{M_1}^\theta S,w \rangle | + 2\varepsilon _\alpha \varepsilon _{\varphi _1}|\langle R’(\overline{\alpha }) \overline{M_1}^\varphi S,w \rangle |\\ & \qquad \quad + \varepsilon _{\theta _1}^2|\langle R(\overline{\alpha }) \overline{M_1}^{\theta \theta } S,w \rangle | + 2\varepsilon _{\theta _1}\varepsilon _{\varphi _1}|\langle R(\overline{\alpha }) \overline{M_1}^{\theta \varphi } S,w \rangle | + \varepsilon _{\varphi _1}^2|\langle R(\overline{\alpha }) \overline{M_1}^{\varphi \varphi } S,w \rangle |\big)\\ & - \frac{(\varepsilon _\alpha +\varepsilon _{\theta _1}+\varepsilon _{\varphi _1})^3}{6},\\ H_P :={}& \langle \overline{M_2}P,w \rangle + \varepsilon _{\theta _2}|\langle \overline{M_2}^\theta P,w \rangle |+\varepsilon _{\varphi _2}|\langle \overline{M_2}^\varphi P,w \rangle |\\ & + \frac{1}{2}\big(\varepsilon _{\theta _2}^2|\langle \overline{M_2}^{\theta \theta } P,w \rangle | + 2\varepsilon _{\theta _2}\varepsilon _{\varphi _2}|\langle \overline{M_2}^{\theta \varphi } P,w \rangle | + \varepsilon _{\varphi _2}^2|\langle \overline{M_2}^{\varphi \varphi } P,w \rangle |\big) + \frac{(\varepsilon _{\theta _2}+\varepsilon _{\varphi _2})^3}{6}, \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

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

(This is a second-order, anisotropic strengthening of [ SY25 ] , Theorem 17, whose penalty is first-order and isotropic with quadratic remainders \(9\varepsilon ^2/2\) and \(2\varepsilon ^2\): here the exact second partials at the center pose are charged with per-axis weights — with multiplicities from the symmetric Hessian table — and only the cubic Lagrange remainders \((\varepsilon _\alpha +\varepsilon _{\theta _1}+\varepsilon _{\varphi _1})^3/6\) and \((\varepsilon _{\theta _2}+\varepsilon _{\varphi _2})^3/6\) are bounded via Lemma 19. On the diagonal \(\varepsilon _\alpha = \varepsilon _{\theta _1} = \dots = \varepsilon \), these are \(\frac{3^3}{6}\varepsilon ^3 = \frac{9\varepsilon ^3}{2}\) and \(\frac{2^3}{6}\varepsilon ^3 = \frac{4\varepsilon ^3}{3}\).)

Proof ▶

See [ SY25 ] , Section 4.2.