All Questions
Tagged with mg.metric-geometry linear-algebra
97 questions
61
votes
11
answers
11k
views
Geometric proof of the Vandermonde determinant?
The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq ...
- 23k
19
votes
3
answers
6k
views
What are the matrices preserving the $\ell^1$-norm?
So I am inspired by unitary matrices which preserve the $\ell^2$-norm of all vectors, so in particular the unit norm vectors. But then I saw that the $\ell^1$-norm of probability vectors is preserved ...
- 391
17
votes
3
answers
1k
views
Prescribing areas of parallelograms (or 2x2 principal minors)
Let $(a_{ij})$ be a $n\times n$ symmetric matrix such that $a_{ij}\geq 0$ for all $i,j$ and $a_{ii}=0$ for all $i$. Under which conditions on the $a_{ij}$'s can one find $n$ vectors $v_1,\ldots,v_n\in{...
- 173
16
votes
1
answer
774
views
Minimizing the excursion of a sum of unit vectors
I have $n$ unit-length vectors $v_i$ in $\mathbb{R}^3$, whose
sum is zero:
$$ v_1 + v_2 + \cdots + v_n = 0 \; .$$
Now I form the closed polygon $P$ in space by placing them head to tail.
So the ...
- 151k
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
15
votes
4
answers
2k
views
More than $n$ approximately orthonormal vectors in $R^n$
This question was asked at math.stackexchange, where it got several upvotes but no answers.
It is impossible to find $n+1$ mutually orthonormal vectors in $R^n$.
However, it is well established that ...
- 1,160
15
votes
3
answers
1k
views
Representation of vectors in $\mathbb{R}^2$ via differences of small vectors.
Is the following fact true?
Let $v_1,\ldots, v_k \in \mathbb{R}^2$, $\|v_i\|\leq 1$, be vectors that add up to zero. Does there exist a permutation $\sigma\in S_k$ and vectors $w_1,\ldots, w_k \...
- 1,284
14
votes
5
answers
2k
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 $...
- 1,513
13
votes
1
answer
329
views
Spectral properties of finite metric sets
Given a finite metric set $S=\{P_1,\dots,P_n\}$, one gets a real symmetric matrix $M=M(S)$
with rows and columns indexed by elements of $S$ by setting
$M_{i,j}=d(P_i,P_j)$.
It is easy to see that $M$...
- 17.9k
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
11
votes
2
answers
797
views
Three half circles on the plane may not meet nicely
Let $H$ denote the union of the northern 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\}$
...
- 2,136
10
votes
2
answers
1k
views
Does the hyperdeterminant calculate a quantity akin to the volume of a parallelepiped?
If $M$ is an $n \times n$ matrix, $|\det(M)|$ is the volume of the $n$-dimensional
parallelepiped spanned by the column vectors of $M$.
...
- 151k
9
votes
2
answers
1k
views
Other norms for lattice reduction techniques (LLL, PSLQ)?
LLL and other lattice reduction techniques (such as PSLQ) try to find a short basis vector relative to the 2-norm, i.e. for a given basis that has $ \varepsilon $ as its shortest vector, $ \varepsilon ...
- 513
8
votes
1
answer
1k
views
Is there an elementary way to show the triangular inequality for this expression ?
Consider the space $X$ of all scalar products on $\mathbb{R}^n$. For a scalar product $s$ and a base $B:=b_1\ldots,b_n$ let $M_{s,B}$ denote the matrix, whose $(i,j)$-th entry is $(s(b_i,b_j))$ . ...
- 11.1k
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
8
votes
0
answers
421
views
Approximate singular value decomposition in Banach spaces
I am interested in generalisations to Banach spaces of the following construction, which relates to the singular value decomposition of a finite-dimensional linear map. If $V$, $W$ are finite-...
- 6,206
8
votes
0
answers
544
views
Maximal set on hypersphere that does not contain pairs of orthogonal vectors
Let R be a region on a hypersphere. Each point A of the hypersphere
is associated with a vector pointing to A and with origin at
the centre of the hypersphere. So let me identify each point with a
...
- 1,207
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
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
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
6
votes
1
answer
761
views
Checking if one polytope is contained in another
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 ...
- 491
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
5
answers
1k
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 ...
- 30.3k
5
votes
3
answers
543
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 ...
5
votes
3
answers
1k
views
Proof that domains of positivity of symmetric nondegenerate bilinear forms are self-dual cones?
Max Koecher (for example, in The Minnesota Notes on Jordan Algebras and Their Applications; new edition: Springer Lecture Notes in Mathematics, number 1710, 1999), defined a domain of positivity for a ...
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
5
votes
2
answers
1k
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 ...
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
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
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
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
5
votes
0
answers
310
views
Biggest (or large) rectangle in a polytope
I need an efficient method to construct a (hyper)rectangle inside a polytope with a lot of dimensions (say $100 < d < 1000$). Ideally I'd want the biggest possible rectangle, but as I don't ...
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
4
votes
1
answer
393
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 ...
- 87
4
votes
1
answer
496
views
Is there a standard measure for how close a matrix is to being a distance metric ?
Suppose I have a square n*n, symmetric matrix with positive elements and zero diagonal.
For this to be considered a proper distance metric between n points, the triangle inequality needs to be ...
- 711
4
votes
2
answers
818
views
Number of independent distances between n points in d-dimensional Euclidean space?
There are $\binom{n}{2}$ distances between $n$ points in $\mathbb{R}^d$. Not all of them can be chosen freely if $n$ exceeds the number $n_d = d + 1$. If $n = n_d$ we obviously have $\binom{d+1}{2}$ ...
- 9,720
4
votes
2
answers
2k
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 $\...
- 143
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
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
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
4
votes
1
answer
232
views
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
4
votes
1
answer
491
views
Generalization of the "double cap conjecture" to a vector space with complex field
The conjecture that I proposed in
Maximal set on hypersphere that does not contain pairs of orthogonal vectors
is in fact known as the "double cap conjecture", as noted by Guillaume Aubrun.
See for ...
- 1,207
4
votes
1
answer
626
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 ...
- 59
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
$$
{\displaystyle d_{\mathrm {H} }(X,Y)=\max \left\{\,\sup _{x\in X, |x|_2=1}\inf ...
- 1,503
4
votes
0
answers
144
views
A Pythagorian inequality characterization of inner-product spaces
Let $(X,\|\cdot\|)$ be a real normed space. For any points $A$ and $B$ in $X$, let $AB:=\|A-B\|$. Suppose that for any points $A$ and $B$ in $X$ and any straight line $\ell\subseteq X$ such that $B\...
- 128k
3
votes
2
answers
5k
views
Volume change under linear transformation
It is well-known, that given a linear transformation $f \colon \mathbb R^n \rightarrow \mathbb R^m$, where $m \ge n$, the $m$-dimensional volume of an image of any measurable subset $S \subseteq \...
3
votes
1
answer
1k
views
Given a distance matrix is there an isometric embedding?
I have distance matrix $D$ that was calculated by some distance (non-Euclidean but satisfying distance requirements). Is there a set of points in some Euclidean space such that it generates matrix of ...
- 33
3
votes
1
answer
187
views
Moment matching on the standard simplex
Let $\vec{\mu}_1, \vec{\mu}_2,\ldots, \vec{\mu}_k \in \Delta^{d-1}$ be $k\ (k\geq 2)$ distinct vectors on the standard simplex, where
$$\Delta^{d-1} = \{\vec{\mu}\in R^{d}:\| \vec{\mu}\|_1 = 1,\mu_j \...
- 1,127
3
votes
3
answers
310
views
measuring $n\ 2$-planes in $\mathbb{R}^{2n}$
Given $n$ vectors $v_1, \ldots, v_n$ in $\mathbb{R}^n$ of course we all know at least one measure for their relative configuration: $|v_1 \wedge\ldots \wedge v_n|$. Now suppose one were given $n$ ...
- 2,274