## 3 Union and Intersection of Thick Ellipsoid Enclosures
### Dikin Ellipsoids for the Intersection of Ellipsoids
#### 3.2.2 Generalization to the Intersection of Ellipsoids with Different Midpoints
: _The thick Dikin ellipsoid_  
\[\left(\xi\right)_{\mathrm{D},k}=\left(\xi\right)_{\mathrm{D},k}\left(\mu_{k},\mathbf{\Gamma}_{\mathrm{D},k}\left[1\,;\,\left(\min_{i\in\{1,\ldots,n\}} \left\{\lambda_{i}\Big{(}\left(\mathbf{Q}_{k}^{\mathrm{O}^{\prime}}\right)^{- 1}\cdot\mathbf{Q}_{k}^{\mathrm{I}}\Big{)}\right\}\right)\right)^{-\frac{1}{2}} \right]\right)\quad\text{with}\quad\mathbf{\Gamma}_{\mathrm{D},k}=\left( \mathbf{Q}_{k}^{\mathrm{I}}\right)^{\frac{1}{2}}. \tag{49}\]  
_represents guaranteed inner and outer bounds of the intersection of two ellipsoids with different midpoints if all parameters are chosen according to Equations (42)-(48)._  
Proof. : The validity of the inner bound is a direct consequence of Lemma 5 in [49], and the construction of the matrix (45) as a guaranteed underapproximation of the domains to be intersected. The outer bound of (49) is represented by the shape matrix  
\[\mathbf{Q}_{k}^{\mathrm{O}}=\alpha_{\mathrm{O},k+1}^{2}\cdot\mathbf{Q}_{k}^{ \mathrm{I}}\ . \tag{50}\]  
It results from Equation (16) after setting \(\mathbf{A}_{k}=\mathbf{I}\), \(\mathbf{\tilde{A}}_{k}=\mathbf{I}\), \(\mathbf{Q}_{k}=\mathbf{Q}_{k}^{\mathrm{O}^{\prime}}\), and \(\mathbf{\mathcal{R}}_{k}=\mathbf{Q}_{k}^{\mathrm{I}}\) which yields  
\[\alpha_{\mathrm{O},k+1}^{-2}\cdot\mathbf{I}\leq\left(\mathbf{Q}_{k}^{\mathrm{ O}^{\prime}}\right)^{-1}\cdot\mathbf{Q}_{k}^{\mathrm{I}}\ . \tag{51}\]  
This inequality is satisfied if  
\[\bar{\mathbf{\mu}}_{k}=\mathbf{\mu}_{1,k}+\mathbf{L}_{k}\cdot\left(\mathbf{\mu}_{2,k}-\bm {\mu}_{1,k}\right) \tag{43}\]0  
where the expression in curly brackets denotes the minimum eigenvalue of the corresponding matrix product. Taking the square root of the inverse of this expression completes the proof of Theorem 5. The alternative solution according to **Step 3*** is given by the thick ellipsoid enclosure  
\[\bar{\mathbf{\mu}}_{k}=\mathbf{\mu}_{1,k}+\mathbf{L}_{k}\cdot\left(\mathbf{\mu}_{2,k}-\bm {\mu}_{1,k}\right) \tag{43}\]1  
which is a direct consequence of Theorem 5 by searching for an ellipsoid parallel to the outer bound that is guaranteed to belong to the interior of the one with the shape matrix \(\mathbf{Q}_{k}^{\mathrm{I}}\). **Remark 8**. : _Due to the fact that the relation_  
\[\bar{\mathbf{\mu}}_{k}=\mathbf{\mu}_{1,k}+\mathbf{L}_{k}\cdot\left(\mathbf{\mu}_{2,k}-\bm {\mu}_{1,k}\right) \tag{43}\]2  
_holds for arbitrary positive definite matrices \(\mathbf{Q}_{k}^{\mathrm{I}}\) and \(\mathbf{Q}_{k}^{\mathrm{O}^{\prime}}\), the ratio between the volumes of the outer and inner ellipsoids in Equations (49) and (53) is identical. Hence, the variant (49) is typically preferred if thick ellipsoids with maximum volume inner enclosures are desired, while (53) provides tighter outer bounds. For ellipsoids with identical midpoints \(\tilde{\mu}_{k}=\mu_{1,k}=\mu_{2,k}\), the ellipsoid of Theorem 3.1 is identical to the one in (41) if the midpoint parameter \(\tilde{\mu}_{k}\) is used instead of the value \(\mathbf{0}\) in (41).