In describing the decomposition of the deformation gradient tensor given by $\mathbf{F}=F_{iJ}=\partial x_i/\partial X_J$, Malvern (1969, Introduction to the mechanics of a continuous medium) mentions that due to the finite nature of the displacements implied in the definition of $F_{iJ}$, it doe...

I want to use Lagrange multipliers to find the minimum separating distance between two ellipsoids, both centered at the origin. To illustrate, we start with both ellipsoids sharing the same center: And want to finish with the closest point that the two ellipsoids may be separated: Attempted sol...

Assume $A \in \textbf{S}^n_{++}$, an ellipsoid centered at the origin given by $$\mathcal{E}_A = \{x\mid x^TA^{-1}x \leq 1\}$$ Then we have $\mathcal{E}_A \subseteq \mathcal{E}_B $ if and only if $B-A \succeq 0$. This is the proposition in the Boyd & Vandenberghe's Convex Optimization (pages 45-4...

In Boyd & Vandenberghe, it is mentioned that the ellipsoid is defined by \begin{equation} \mathcal{E} = \left\{ x \in \mathbb R^n \mid (x-x_c)^T P^{-1} (x-x_c) \leq 1 \right\} \end{equation} where $P$ is positive definite. My questions are: How to transform it to the normed (in)equation? How ...

Given a convex body $K$ and an ellipsoid of minimal volume which contains $K$, find the maximal ellipsoid contained in $K$. I have tried to multiply the matrix by 4 (since the eigenvalues are the reciprocals of the squares of the semi-axes, but I don't get the maximal ellipsoid. If you need to...

I should start by saying that I haven't done algebra for very long time. I recently have some work related to algebra, so I need some help to speedup. I went through a theorem in the book stating the relationship between ellipsoid's radii and the eigenvalues. So the ellipsoid is defined as foll...

think of an ellipsoid in the n-dimensional space defined by $$(x-\mu)'A(x-\mu)=1.$$ I was calculating the volumes of n-dimensional ellipsoids like the one from above for a while, which is straightforward once the eigenvalues of matrix $A$ are retrieved. The volume $V$ is then given by (using the...

Specifically, given a positive definite matrix $A \in \mathbb{R}^{n \times n}$, how can I efficiently generate points $x \in \mathbb{R}^n$ that satisfy $x^TAx \leq 1$? I know how to do this when the matrix is just the identity, because then this question reduces to simply generating uniformly fro...

I need your expertise in solving the following the follow problem: Given a convex body P, which is given implicitly by an oracle (either membership or separation), the objective is to find a minimum enclosing ellipsoid using the ellipsoid method. To achieve such task, we need to formulate the p...

Let $\mathcal{S}$ be an ellipsoid in $\mathbb{R}^n$ and let $U\in\mathbb{R}^{n\times n}$ be a unitary matrix. Is it true that $U\mathcal{S}=\{y| \quad \exists x \in\mathcal{S} \quad s.t. \quad y=Ux\}$ is also an ellipsoid?

How can the packing density of a set of congruent ellipsoids be calculated? I'm dealing with prolate spheroids so technically I do not need the general answer for ellipsoids, but my abstract mind loves more general answers. If calculating an exact value is too difficult, an estimate which is accu...

How do I compute the volume of the intersection of two $n$-dimensional ellipsoids? Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid $$E(c,A)=\{x|(A(x-c),x-c)<1\}$$ where $(\cdot,\cdot)$ if the dot product. Then $$\mathrm{vol}(E(c,A))=\frac{u...

Show that an ellipsoid $$\{x\in \mathbb{R}^n \ : \ x^TAx+2b^Tx+c\le 0\},$$ where $A\in \mathbb{S}^n_+$, is a convex set.

Let $E$ be an ellipsoid in $\mathbb{R}^d$ defined by $$\sum_{i=1}^d \frac{x_i^2}{a_i^2}=1$$ Is there a formula to express the mean width (or an approximation of the mean width) of $E$ in term of the lengths $a_i$ of the (semi-principal) axis? The width in the direction of the principal axis are $...

I have an ellipsoid centered at the origin. Assume $a,b,c$ are expressed in millimeters. Say I want to cover it with a uniform coat/layer that is $d$ millimeters thick (uniformly). I just realized that in the general case, the new body/solid is not an ellipsoid. I wonder: How can I calculate th...