# Tagged Questions

63 views

### Are there a group of mappings from (n-1)-dim space to an (n-1)-sphere guaranteeing the orthogonality of images?

Hello, everyone. As we know that in an $n$-dimensional Euclidean space $\mathbb{R}^n$, there exists a continuous bijective mapping from a subset $V^{n-1}\subseteq\mathbb{R}^{n-1}$ to a unit ...
987 views

### Geometric interpretations of matrix inverses

$A$ is an invertible $n \times n$ matrix. Interpret each row of $A$ as a point in $\mathbb{R}^n$; then these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through each point ...
456 views

### How to efficiently compute the generalized cross product?

It's possible to extend the well known cross product between two vectors in $\mathbb{R}^3$ to $n-1$ vectors in $\mathbb{R}^n$. Let $\vec{v_1}, \vec{v_2}, \dots, \vec{v}_{n-1} \in \mathbb{R}^n$ and ...
464 views

### Three half circles on the plane may not meet nicely

Let $H$ denote the union of the northen hemisphere of the unit circle $S^{1}$ with the interval $[-1,1]$ on the $x$-axis. That is, $H=\{(x,\sqrt{1-x^{2}}):-1\le x\le 1\}\cup\{(x,0):-1\le x\le 1\}$ ...
263 views

### Can an ellipsoid be moved freely inside another ellipsoid?

An origin centric ellipsoid is defined by any positive semi-definite $n$ by $n$ matrix $X$, by taking all vectors $v$ such that $v^tXv\leq1$. Call two origin centric ellipsoid equivalent if one can be ...
510 views

### Finding an axis-aligned ellipsoid of minimal volume which contains a given ellipsoid

A friend asked me to post the following question. He's not an MO user and felt it would be better received if asked by someone who was already known to the community. This is not my area, but I'll do ...
1k views

### Prescribing areas of parallelograms (or 2x2 principal minors)

Let $(a_{ij})$ be a $n\times n$ symmetric matrix such that $a_{ij}\geq 0$ for all $i,j$ and $a_{ii}=0$ for all $i$. Under which conditions on the $a_{ij}$'s can one find $n$ vectors ...
820 views

### How far is a set of vectors from being orthogonal?

Given some vectors, how many dimensions do you need to add (to their span) before you can find some mutually orthogonal vectors that project down to the original ones? Or, more formally... Suppose ...
209 views

### Checking if one polytope is contained in another

Hi, I have two sets of inequalities, say, $Ax \leq 0$ and $Bx \leq 0$. I would like to know if they both define the same polytope. Or, even, whether one is contained in the other. At the moment I am ...
344 views

### Finding a hyperplane that splits a convex polytope evenly

Say we have a convex polytope in standard form: \begin{equation*} \begin{array}{rl} \mathbf{A}\mathbf{x} = \mathbf{b} \\\\ \mathbf{x} \ge 0 \end{array} \end{equation*} Are there any known methods ...
157 views

### Looking for a simple proof that the generalized disc is bounded

So let us define the generalized disc of degree $n$ as $$\mathbb{D}_n:=\{w\in M_{n\times n}(\mathbb{C}):w=w^t, I_n-w\overline{w}>0\}.$$ For a Hermitian matrix $A$, the notation $A>0$ means ...
393 views

### Are these vectors in the non-negative orthant of some $R^K$?

Given three sets of vectors $S_i=\left(V_{i1},V_{i2},\ldots,V_{in}\right),i=1,2,3$, s.t. the vectors within a set are pair-wise orthogonal and $V_{ij}\cdot V_{kl}\geq 0$ $\forall i,j,k,l$. Also, ...
332 views

### Feasible space of SDP

Typically the non-empty feasible space of a SDP has some curved boundary which is why the feasible space has infinitely many extreme points. Is it ever possible to have a SDP whose non-empty feasible ...
390 views

### Embed the intersection of an n-dimensional unit $L_1$ sphere and a hyperplane into an (n-1)-dimensional unit $L_1$ sphere.

In $\mathbb{R}^n$, given an unit $L_1$ sphere $\mathcal{B}_n: |x_1|+|x_2|+\ldots+|x_n|\leq 1$ and a hyperplane $\mathcal{P}: a_1x_1+a_2x_2+\ldots+a_nx_n=0$. Does there always exist a rotation such ...
494 views

### Find the point on the Stiefel Manifold that is closest to a matrix

I don't have much background on high-dimensional geometry, so I dare to ask it. For a given point in $x\in\mathbb{R}^n$, assume that I want to find the point on the unit sphere that is closest to the ...
295 views

### How do maximum norms relatively change in Euclidean translations

Let $Q$ be the cube $[-1,1]^{3}$ and $\pi$ be a plane in $\mathbb{R}^{3}$ that contains the origin but doesn't contain any vertex of $Q$. Suppose that $A$ is an invertible linear transformation from ...
510 views

### Sequential sampling of Gaussian and von Mises-Fisher Random Variable

I don't find any article discussing this problem, so I dare to ask it. Suppose we are dealing with a data $x_0 \in \mathbb{R}$ and a function $f:\mathbb{R} \to \mathbb{R}$. Say we repeatedly apply ...
336 views

### Connections between a polytope's symmetry group and the existence of periodic orbits

Given an $n$-dimensional convex polytope $P$, one may set into motion a point-mass, starting on one of the facets of $P$, which travels along a straight trajectory inside $P$ except on collision with ...
1k views

### Matrix approximation

Let A be an $m\times n$ matrix and $k$ be an integer. Assume that $A$ is non-negative. We want to find a scalar $\epsilon$ and an $n\times n$ matrix $B$ such that $A\leq A(\epsilon I + B)$ (where ...
Let K be an ordered field. Define the n-sphere: $$S^n(K) := \{ (x_1,x_2,\dots,x_n+1) \in K^{n+1} \mid \sum_{i=1}^{n+1} x_i^2 = 1 \}$$ A set of vectors $v_1, v_2, \dots, v_r \in S^n(K)$ is ...
The question pertains to a polytope that is generated by the intersection of an affine subspace with a hypercube in $p$ dimensions. The affine subspace is given by: $X \mbox{ u} = y$ where $u$ ...