All Questions
Tagged with mg.metric-geometry linear-algebra
97 questions
0
votes
1
answer
114
views
Geometric interpretation of a Grammian-like function
Let $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ and consider the following function $f : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$:
$$
f(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|\|\mathbf{w}...
- 6,351
2
votes
1
answer
312
views
Question on a vector inequality
Is it true that
$$
\min\left( \begin{aligned}
&\|\mathbf{u}\| + \|\mathbf{v}\| - \|\mathbf{u} + \mathbf{v}\|, \\
&\|\mathbf{u}\| + \|\mathbf{w}\| - \|\mathbf{u} + \mathbf{w}\|, \\
&\|\...
- 171
2
votes
0
answers
38
views
Constructing an $n$-simplex at the border of a $n$-ball by orthogonal hyperplanes
I want to construct an $n$-simplex the following way:
Choose $n$ vectors in the boundary of an $n$ dimensional ball, which are forming an $(n-1)$-simplex together.
Place the orthogonal affine $n-1$-...
1
vote
0
answers
27
views
Seeking Help with Classifying Polygons: Waterholes and Airpockets in 2D Space
I am currently in the process of writing software and have encountered a mathematical problem. Perhaps there are some experts here who are familiar with this. It involves the classification of ...
- 11
1
vote
1
answer
132
views
Can I find $n$ points on the boundary of an $n$-dimensional ball with certain properties?
My problem is the following: I want to construct $n$ rays all starting at a point $v$ that is not in the $n$-dimensional ball around $0$ such that the following is true:
The $n$-dimensional ball is a ...
0
votes
0
answers
96
views
When can a point be reconstructed from relative angle measurements?
Given a set of points $p_1,\dots,p_n$ in $\mathbb{R}^d$ and a target point $x\in\mathbb{R}^d$, I measure all the angles between all pairs of points and the target point. In other words, I have the ...
- 427
2
votes
1
answer
237
views
Geometric interpretation of trace of a linear operator
This question is really an addendum to Geometric interpretation of trace
There is a nice account of the trace in Chris Doran's thesis here: http://geometry.mrao.cam.ac.uk/wp-content/uploads/2015/02/...
- 21
0
votes
0
answers
28
views
Example of a matrix -HDH that is not PSD (with non-euclidean distances D)
It's widely known that, given a matrix of squared Euclidean distances, $\mathbf{D}_{ij} = \| \mathbf{X}_i - \mathbf{X}_j \|^2$, and the centering matrix $\mathbf{H} = \mathbf{I} - \dfrac{1}{n}11^T$, ...
- 101
5
votes
0
answers
137
views
Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ are needed to uniquely determine all inner products
Given a collection of vectors $x_1,\ldots,x_k$, which inner products $\langle x_i,x_j\rangle$ need to be known to uniquely determine all inner products? I'll begin with the specific case I am ...
- 351
1
vote
0
answers
43
views
Intersection of unit-norm vectors with a large sum in high dimensions with a spherical cap
Let $d$ and $n$ be integers. For $i \in \lbrace 1,\dots,n \rbrace$ let $x_i \in \mathbb{R}^d$ be a vector such that $\lVert x \rVert=1 $. For a fixed $1/2 < \alpha \leq 1$, assume we have $\lVert \...
- 139
1
vote
1
answer
184
views
Average distance between points of lower dimensional simplices in $\mathbb R^n$
Notation: By a simplex, we mean the convex hull of a finite set of distinct points in $\mathbb R^n$, which are called the vertices of the simplex. $\mathcal H^n$ will denote the $n$-dimensional ...
- 6,155
6
votes
2
answers
539
views
Conditions for including cones
Consider $N$ $n$-dimensional vectors, where the angle between any two vectors is acute and their starting point is at the origin. Can we rotate these vectors together so that the coordinate components ...
- 61
5
votes
1
answer
167
views
What structure is preserved by pseudo-homeomorphisms of pseudo-Euclidean spaces?
Let us recall that for integer numbers $t,s\ge 0$ the pseudo-Euclidean space $\mathbb R^{t,s}$ is the vector space $\mathbb R^{t+s}$ endowed with the quadratic form $q_{t,s}:\mathbb R^{t+s}\to\mathbb ...
- 41.8k
4
votes
1
answer
232
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Is the affine geometry a geometry of proportions?
Given any linear space $L$ over an ordered field $F$, consider the equiproportion relation $${\sim}=\{((x,y,z),(a,b,c))\in L^3\times L^3: \exists t\in[0,1]\subseteq F\;(y{-}x=t(z{-}x)\wedge b{-}a=t(c{-...
- 41.8k
2
votes
0
answers
233
views
Do you know this formula for the scalar product in barycentric coordinates?
I've found a formula for a scalar product in barycentric coordinates which I think is pretty cool. I hope that it's new. Is it?
Suppose that you have points $x_1,\dots,x_n$ sitting in general position ...
- 1,048
1
vote
0
answers
34
views
Maximum number of concurrencies for $J\cdot L$ hyperplanes in $\mathbb{R}^{J-1}$
I have $J\cdot L$ hyperplanes in $\mathbb{R}^{J-1}$ and want to prove that there cannot be more than $L$ points where $J$ hyperplanes intersect simultaneously (aka. concurrencies).
I suspect that the ...
1
vote
0
answers
109
views
Relation between the dimension of vector spaces and dimension of the space [closed]
Let $A \in \mathrm{GL}(d, \mathbb{R})$ be an irreducible matrix. Assume that $\{V_{n}\}_{n\in \mathbb{N}}$ is a non-zero proper subspace $\mathbb{R}^d$ with dimension $t<d,$ such that $AV_{n}=V_{n+...
- 133
1
vote
1
answer
81
views
Generate an ordered set of mostly orthogonal vectors $\{x_i\}$ where $x_i \cdot x_j =0$ iff $\lvert i-j\rvert >m$
I am wondering if there is a way to formulate or generate a matrix $X \in R^{n\times n}$ whose column vectors $\{x_1,\dotsc,x_i,\dotsc,x_n\}$ are such that $x_i$ and $x_j$ are orthogonal iff $\lvert i-...
- 433
1
vote
1
answer
137
views
Relationships among lattices U14, C2xG23, A15+ and their Delaunay polytopes
Do you have any references explaining the relationships among the lattices U14, C2xG23 aka Q14, and A15+?
Do you have any references explaining the relationships among these lattices and the 7D ...
- 53
0
votes
1
answer
445
views
Standard Gram matrices for lattices
I would like to define standard Gram matrices, and use them to help me understand the symmetries of lattices.
I define "standard Gram matrix" as the Gram matrix g that minimizes the ...
- 53
2
votes
2
answers
248
views
On an angle distribution of a random linear subspace of a given dimension
$\newcommand\R{\mathbb R}$ Let $u$ be a fixed unit vector in $\R^n$, and let $\Pi_u$ be the hyperplane in $\R^n$ with normal vector $u$. Let $B$ be the (say open) unit ball in $\R^n$ centered at the ...
- 128k
1
vote
0
answers
70
views
Distance to set defined as subzero level set of a continuous function
I am searching for strategies on how to prove/disprove that scalar functions "capture" the distance to the subzero level set of the same function. (Or what topics to study to become better ...
1
vote
1
answer
428
views
Covering number in the space of symmetric matrices
Let $S_n(\mathbb{R})$ be the set of symmetric matrices of size $n \times n$. Note $\|\Theta\|_{0}$ the number of nonzero elements of a matrix $\Theta$ and $\|\cdot\|_F$ the Froebenius norm. Consider ...
- 407
1
vote
1
answer
319
views
Are there any applications of linear algebra over the complex numbers, where the role of complex conjugation is replaced with the trivial involution?
The complex inner product $\langle u, v \rangle$ is a special case of a sesquilinear form over a field. Its definition is $\langle u, v \rangle = \sum_{i} u_i \overline{v_i}$. There is clearly the ...
- 4,943
4
votes
3
answers
246
views
When does a finite metric induce a matrix norm?
If I have a metric $d(\cdot,\cdot)$ on the set $\{1,\dots,n\}$, are there well-known necessary or sufficient conditions for the existence of a matrix norm $Q$ that induces that metric on the unit ...
- 4,049
7
votes
0
answers
254
views
Set of unit vectors such that among any three there is an orthogonal pair
I was fascinated by the solutions of Problem 8 of the IMC 2021 contest, which can be summarized as:
Theorem 1. Let $v_1,\dotsc,v_N$ be distinct unit vectors in $\mathbb{R}^n$ such that among any three ...
- 105k
1
vote
0
answers
152
views
Reference request: a class of matrices leading to interesting metric geometry
For $0 \le A \in GL(n,\mathbb{R})$, let $Aw = \Delta(A)$, where $\Delta$ denotes the map taking a matrix to a vector of its diagonal entries and/or forming a diagonal matrix from a vector, according ...
- 15.4k
5
votes
2
answers
134
views
Is there a non-orthogonal linear deformation of a polytope that preserves edge-lengths and vertex-origin-distances?
Is there a polytope $P\subset\Bbb R^d$ (convex hull of finitely many points, not contained in a proper affine subspace), and a linear, but non-orthogonal transformation $T\in\mathrm{GL}(\Bbb R^d)\...
- 13.6k
16
votes
1
answer
537
views
Balls in Hilbert space
I recently noticed an interesting fact which leads to a perhaps difficult question. If $n$ is a natural number, let $k_n$ be the smallest number $k$ such that an open ball of radius $k$ in a real ...
- 2,049
5
votes
0
answers
144
views
Do products of distance functions separate points?
Let $(X,d)$ be a metric space without isolated points and of diameter $1$. Let $Y=\{y_m\}_{m=1}^{\infty}$ be a dense subset of $X$.
Define $g_0\equiv 1$, and for $m>0$ let $g_m=d(\cdot,y_1)\dotsm d(...
- 5,529
2
votes
0
answers
46
views
signatures of quasi-gram matrices
Suppose I have a finite subset $\mathcal{M}$ of a Banach space $B$ $\mathcal{M}=p_1, \dots, p_n,$ and I create the following ``Gram'' matrix $G_{\mathcal{M}}:$
$$g_{ij} = \frac{\|p_i\|^2 + \|p_j\|^2 ...
- 96.4k
4
votes
0
answers
223
views
$\epsilon$-net under Hausdorff distance
Consider linear subspaces of $\mathbb{R}^n$. For two subspaces $X$ and $Y$, we define their Hausdorff distance as
$$
{\displaystyle d_{\mathrm {H} }(X,Y)=\max \left\{\,\sup _{x\in X, |x|_2=1}\inf ...
- 1,503
4
votes
0
answers
144
views
A Pythagorian inequality characterization of inner-product spaces
Let $(X,\|\cdot\|)$ be a real normed space. For any points $A$ and $B$ in $X$, let $AB:=\|A-B\|$. Suppose that for any points $A$ and $B$ in $X$ and any straight line $\ell\subseteq X$ such that $B\...
- 128k
2
votes
0
answers
51
views
Find Line through the origin which minimizes the squared perpendicular distances to given points [closed]
Given a set of Points $p_i\in \Re^d$, I'm looking for the vector $x\in\Re^d$ with $\| x \|=1$ along a line so that minimizes $$\sum_{i}(\|p_i\|^2 - \langle p_i, x\rangle^2)$$
According to the first ...
- 21
1
vote
0
answers
1k
views
How to project a matrix to a unitary matrix?
Given a nonzero vector $v \in \mathbb{R}^n$, we all know that it's projection onto the unit $\ell_2$ ball is just $\frac{v}{\|v\|}$. Let $X$ be some nonzero $n \times n$ matrix. What is the projection ...
- 1,703
6
votes
0
answers
217
views
Is this function embeddable in Euclidean space?
Let $X = \{v_1,\ldots,v_n\}$ be a set of vectors non-zero vectors $v_i \ge 0$ and such that the vectors are pairwise linear independent. Define a function on this set $X$:
$$d(v,w) = 1-\frac{2 \...
user6671
5
votes
0
answers
350
views
How to calculate the volume of a parallelepiped in a normed space?
Let $E$ be a real normed space, and let $v_1,...,v_n\in E$ be linearly independent. The parallelepiped defined by these vectors is $P=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Since $E$ is a ...
- 5,529
4
votes
1
answer
1k
views
What is the average area of the shadow of a convex shape taken over all possible orientations?
If we take a sphere, its surface area is exactly 4 times the area of its shadow. How can it be generalised for any convex shape?
- 343
1
vote
1
answer
95
views
A question on a special "metric"
Suppose we have a function $F: [a,b]^n \to \mathcal{M}_{n \times n }(\mathbb{R})$ where $\mathcal{M}_{n \times n }(\mathbb{R})$ is the space of $n \times n$ real matrices, a compact set $B \subset \...
- 113
8
votes
0
answers
194
views
Geometric mean of three or more positive definite matrices
The geometric mean of two positive definite (Hermitian) matrices of same size is defined by $$A\natural B := A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$$equivalently,
$$A\natural B =(BA^{-1})^{1/2}A=A(A^...
- 13.4k
8
votes
0
answers
233
views
A conjecture on simplex
Let $A_0A_1...A_n$ be a simplex in $\Bbb E^n.$ Let $B_{ij}$ be a point on the edge $A_iA_j,\ 0\le i\not=j\le n.$
Denote by $\beta_i$ the hyperplane passing through the points $B_{i0},$ $B_{i1},$ $B_{...
- 705
2
votes
2
answers
273
views
Johnson-Lindenstrauss Lemma on $S^{d-1}$
Consider the Johnson-Lindenstrauss lemma in the case where we can assume the $n$ input points $x_i$ in $\mathbb{R}^d$ are actually located on the sphere
$$S^{d-1}(r):=\{u=(u_1,\ldots,u_{d}): u_1^2+\...
- 10.4k
19
votes
3
answers
6k
views
What are the matrices preserving the $\ell^1$-norm?
So I am inspired by unitary matrices which preserve the $\ell^2$-norm of all vectors, so in particular the unit norm vectors. But then I saw that the $\ell^1$-norm of probability vectors is preserved ...
- 391
2
votes
0
answers
48
views
Minimization of the volume of the image of space-filling convex polytopes under similarities
Suppose $A:\mathbb{R}^n \to \mathbb{R}^n$ is a similarity, given by $A(x) = \lambda Ox$, where $\lambda > 1$ and $O$ is an orthogonal matrix (i.e., $A$ is a particular loxodromic repelling ...
- 21
7
votes
2
answers
1k
views
Maximum average Euclidean distance between $n$ points in $[-1,1]^n$
For my research I have designed a metric that is based on the average Euclidean distance between $n$ points in the $n$-dimensional hypercube $[-1,1]^n$. However, I have a hard time finding the maximal ...
- 71
4
votes
2
answers
492
views
The Aleksandrov-Fenchel inequality of mixed discriminants for Hermitian matrices
Suppose $A,A_1,\ldots,A_{n-2}$ (resp. $B$) are (resp. is) real positive-definite (resp. arbitrary) symmetric $n\times n$ matrices and denote by $D(\cdot,\ldots,\cdot)$ the mixed discriminant. We have ...
- 593
3
votes
0
answers
81
views
Iterated crossproducts
Let $S_0$ be a set of vectors in $\mathbb{R}^n$. Now iteratively define $S_{j+1}$ by taking all possible crossproducts of vectors in $S_j$ and normalizing such that the vector of maximum norm has unit ...
- 101
4
votes
2
answers
209
views
Geometrical interpretation of pictures transforms and other "high dimensional everyday objects"
During the preparation of a general audience talk on why mathematicians use dimensions higher than three (or four) even for concrete applications, I came up with the following enjoyable observation : ...
- 2,135
13
votes
2
answers
794
views
Distance of vectors versus distance of their difference vectors
For any given $x \in \mathbb{R}^n$, let $\nabla{x} \in \mathbb{R}^{n \choose 2}$ be the vector whose $\{i,j\}$-th entry is $|x_i-x_j|$. I think the following claim is true.
Claim. If $f, g \in \...
- 519
2
votes
0
answers
92
views
Estimating the size of a subset of $\mathbb{R}^N$
This concrete geometric question has arisen out of the problem of counting arithmetic functions with a particular property. The details of the relationship between the counting procedure and this ...
- 2,480