2
votes
0answers
62 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 ...
7
votes
1answer
941 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 ...
3
votes
1answer
446 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 ...
8
votes
2answers
463 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, $H=\{(x,\sqrt{1-x^{2}}):-1\le x\le 1\}\cup\{(x,0):-1\le x\le 1\}$ ...
3
votes
1answer
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 ...
5
votes
5answers
504 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 ...
16
votes
3answers
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 ...
14
votes
5answers
814 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 ...
4
votes
1answer
206 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 ...
5
votes
3answers
340 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 ...
2
votes
2answers
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 ...
1
vote
0answers
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, ...
2
votes
1answer
331 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 ...
4
votes
1answer
388 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 ...
4
votes
2answers
491 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 ...
1
vote
1answer
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 ...
1
vote
1answer
505 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 ...
3
votes
1answer
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 ...
2
votes
1answer
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 ...
6
votes
2answers
690 views

Spheres over rational numbers and other fields

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 ...
1
vote
1answer
374 views

Is the direction of the longest line of a polytope unique?

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 $u$ ...