All Questions
4,825 questions
0
votes
0
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67
views
Projection of a gaussian random vector onto a convex body
Let $K \subset \mathbb{R}^n$ denote a convex body. Let $\Pi_K$ denote the projection onto $K$,
$$
\Pi_K(y) = \mathrm{arg\,min}_{x \in K} \|y - x\|,
$$
where $\|\cdot\|$ denotes the usual Euclidean ...
- 460
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 $f : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$:
$$
f(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|\|\mathbf{w}...
- 6,351
1
vote
0
answers
33
views
Obtaining the geodesic extension property by embedding in a larger space
Suppose $(X,d)$ is a Hadamard space. By considering basic examples like a compact interval in $\mathbb{R}$ or a closed unit ball in Hilbert space, $X$ need not have the geodesic extension property (...
- 163
0
votes
0
answers
21
views
Largest inscribed parallelepiped of the convex set defined by partial sum of Fourier series
Let $\mathcal{X}$ be the set consisting of all $(2n+1)$-dimensional real vectors $\mathbf{x}=\left( a_0,a_1,\ldots,a_n,b_1,\ldots,b_n\right)^{\intercal}$ satisfying
$$
\left| f_{\mathbf{x}}(t) \right|...
- 550
0
votes
0
answers
106
views
Upper bounds for minimum angle
What are the latest and best results on the asymptotic upper bound for the minimum angle between any pair of rays among $n$ rays in $\mathbb{R}^3$?
Any helpful answer would be appreciated. Thank you!
- 61
7
votes
0
answers
315
views
Sandwiching ellipses between planar convex bodies
Let $K$ and $L$ be planar convex bodies which are not ellipses. Does there exist an affine image $K'$ of $K$ such that
$K' \subset L$
No ellipse $E$ satisfies $K' \subset E \subset L$
I am also ...
- 4,653
1
vote
0
answers
103
views
Low dimensional symmetric Euclidean immersions of the Klein quartic
I was wondering what the lowest Euclidean dimension the Klein quartic has a faithful symmetric immersion, that is an immersion such that all of its automorphisms* are distinct symmetries of the ...
- 255
7
votes
1
answer
179
views
More on the Gram matrix of $6$ unit vectors in $\Bbb R^3$
Let $G=(g_{ij}\colon i,j=1,\dots,6)$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$. Let
$$u:=\sum_{1\le i<j\le 6}g_{ij}^2,\quad v:=\sum_{1\le i<j<k\le 6}g_{ij}g_{ik}g_{jk}.$$...
- 128k
4
votes
1
answer
96
views
On the Gram matrix of $6$ unit vectors in $\Bbb R^3$
Let $G$ be the $6\times6$ Gram matrix of $6$ unit vectors in $\Bbb R^3$.
Can the mean of the squares of the off-diagonal entries of $G$ be $<1/5$?
Remark 1: A numerical experiment suggests that $...
- 128k
1
vote
1
answer
111
views
Question on constraints
Does there exist any set of 6 real numbers $a_{ij}$ for $1 \leq i < j \leq 4$ satisfying the following conditions:
\begin{aligned}
&0 \le a_{ij} \le \pi, \\
&a_{ik} + a_{jk} > a_{ij}, \\
...
- 61
1
vote
0
answers
71
views
Is the circumcenter Lipschitz on large convex sets in hyperbolic space?
Given a uniquely geodesic metric space $X$, let $\mathcal K(X)$ denote the metric space of compact, convex subsets of $X$ equipped with the Hausdorff distance. Given $K \in \mathcal K(X)$, let $c(K)$ ...
- 498
0
votes
2
answers
134
views
What is the smallest area of a central section of the unit hypercube?
Let $\mathcal{U} \subseteq \mathbb{R}^n$ denote the unit hypercube i.e. $\mathcal{U} = [0,1]^n$, and assume that for some $d \in \mathbb{R}^n$ one denotes by
$$
\mathcal{H} = \left\{x \in \mathbb{R}^n ...
- 251
4
votes
0
answers
66
views
Convergence of graph geodesics to geodesics on metric spaces
Let $(X,d)$ be a compact length space metric space $\mathbb{X}_{\delta}$ be a $\delta$-packing on $X$ and, for every $k\in \mathbb{N}_+$, let $G_{k,\delta}=(\mathbb{X}_{\delta},\mathcal{E}_k,W_k)$ ...
- 364
0
votes
0
answers
24
views
Is there a log-concave distribution not spherical symmetric s.t $ \langle X, \theta \rangle$ is almost normal for all directions $\theta$?
Klartag's results indicate that for a log-concave isotropic random vector, with high probability over $\theta$, $\langle X, \theta \rangle$ is close to a normal distribution.
It is known that for the ...
- 11
2
votes
1
answer
312
views
Question on a vector inequality
Is it true that
$$
\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
70
views
Lipschitz continuity of orthogonal projection with respect to the Hausdorff distance
Let $x_0 \in \mathbb R^n$, and let $\mathcal K$ denote the set of compact convex subsets of $\mathbb R^n$ equipped with the Hausdorff metric. Consider the map $f: K \mapsto \Pi_K x_0$, where $\Pi_K$ ...
- 498
1
vote
0
answers
162
views
A certain circle formed by perpendiculars
If six points are chosen, two points on each side of a triangle, such that they have the same ratio of distances to vertices, then the perpendicular lines through those points meet at six concyclic ...
- 1,109
5
votes
1
answer
247
views
Question on the exact largest minimum angle
Could anyone help find the EXACT largest minimum angle between any pair of lines among 5 lines passing through the origin in $\mathbb{R}^3$? Additionally, what is the exact largest minimum angle ...
- 61
6
votes
1
answer
241
views
Reference request: acceleration/curvature of curve in metric space
Let $(X,d)$ be a metric space. Given a continuous curve $\gamma_t : [0,1] \rightarrow X$, the metric speed is defined by $$ |\gamma_t^\prime | := \lim_{s\rightarrow t} \frac{d(\gamma_s, \gamma_t)}{|t-...
- 660
5
votes
3
answers
286
views
On a metrized $n$-dimensional manifold $X$, does every $x \in X$ have a small ball $B_\delta(x)$ that is homeomorphic to $\mathbb R^n$?
Suppose that $X$ is an $n$-dimensional topological manifold that is also metrizable, and hence equipped with some metric that induces the topology.
For every point $x \in X$, let $B_\delta(x)$ be the ...
- 5,327
20
votes
2
answers
3k
views
Can a 2-sphere be squashed flat?
Does there exist a function $f:\Bbb{S}^2\rightarrow\Bbb{R}^2$ which preserves the length of every rectifiable curve? That is, can a sphere be crushed flat without tears? Of course, this is a Nash-...
- 420
2
votes
1
answer
147
views
Are there polytopes with precisely two realizations?
A convex polytope is projectively unique if it has a unique realization up to projective transformations. Such polytopes are somewhat mysterious but still well-studied. Examples are simplices, the ...
- 13.6k
2
votes
0
answers
48
views
Maximum coverage of an orthogonal polygon using $k$ rectangles
I have an orthogonal polygon (all edges are horizontal or vertical) which is convex (no holes in any row of column of the polygon).
I would like to cover as much as possible of this orthogonal polygon ...
- 21
5
votes
1
answer
536
views
Twin circles in a quadrilateral
The circumcenters of the four triangles of a complete quadrilateral along with the two points of completion form two congruent circles (in black).
Surely this must've been done before - what's the ...
- 1,109
2
votes
0
answers
82
views
Is isoperimetric hypersurface unique up to homeomorphism?
Is there a Riemannian structure on $\mathbb{R}^n $with two non homeomorphic compact hypersurfaces $M,N$ such that both satisfy the isoperimetric inequality. I precisely meanthe following:
$$\...
- 356
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
1
answer
118
views
Contraction and consensus on Hadamard manifolds
Let $\mathcal M$ be a Hadamard manifold and $\{x_i\}_{i=1}^n\subseteq{\cal M}$ be $n$ points. Define $\{y_i\}_{i=1}^n$ as the weighted Fréchet means:
$$
y_i=\arg\min_{y\in\mathcal M}\sum_jw_{ij}d^2(y,...
- 125
0
votes
0
answers
119
views
Boundedness of 2 times the unit ball
Suppose that $X$ is a topological vector space where the topology is given by a metric $d$ on $X$. Assuming that the unit ball
$$
B(0, 1) := \{x \in X : d(0, x) < 1\} \neq X,
$$
is it necessarily ...
3
votes
0
answers
61
views
For which lattices L does the cluster of Voronoi regions abutting that of the origin have a lattice tiling of euclidean space?
Let L be a n-dimensional lattice (a discrete cocompact subgroup of n-space).
Let V0 denote the Voronoi region of the origin, and let C denote the union of V0 with all the Voronoi regions that share a ...
- 2,858
0
votes
0
answers
37
views
Does smallness of Gromov-Hausdorff distance on scale 2 imply smallness on GH distance on scale 1?
Let $(M,g)$ be a Riemannian manifold and $C(Y)$ be a metric cone over $Y$. Let $B_r$ denote the geodesic ball of radius $r$ centered at a fixed point $x$ in $M$ and $C_r$ denote the metric ball of ...
- 151
8
votes
0
answers
149
views
Do the $\ell^{\infty}$ and $\ell^1$ norms yield minimal doubling constants amongst all norms on $\mathbb{R}^n$?
Setting:
Let $X:=\mathbb{R}^n$ for some positive integer $n$. For each $1\le p\le \infty$ let $d_p$ denote the metric induced by the $\ell^p_n$ norm thereon.
Note that, the doubling constant of a ...
- 5,405
0
votes
0
answers
78
views
Coordinates of the centers of the insphere and circumsphere
Suppose we are working in $\mathbb{R}^3$ space and we have four non-coplanar and non-collinear points, $(x_a, y_a, z_a)$, $(x_b, y_b, z_b)$, $(x_c,y_c, z_c)$, and $(x_d, y_d, z_d)$. How does one ...
- 1,109
2
votes
0
answers
113
views
Another Butterfly theorem — Conway like circle
Have You seen these result as follows before?
In Figure 1: $AA'=BB'=tAB$; $CC'=DD'=tCD$, where t is real number then $ABCD$ is a cyclic quadrilateral iff $A'B'C'D'$ is a cyclic quadrilateral.
In the ...
- 1,776
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
46
views
Can either pair of opposite sides of an arbitrary parallelogram be brought into coincidence isometrically in 3-space?
Let P denote any nondegenerate planar parallogram, and let A and B be either pair of its opposite edges.
Does there always exist a continuous family of locally-isometric mappings ht of P into 3-space, ...
- 2,858
2
votes
0
answers
44
views
Under what conditions do distances from pivot points uniquely identify a point on a manifold?
Let $X$ be a smooth manifold of dimension $n$ equipped with a Riemannian metric.
Suppose that $x_1, \dots, x_m$ are pivot points on that manifold. We consider the distance functions
$$
f_i(x) = d( x_i,...
- 5,327
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 ...
4
votes
0
answers
87
views
Statistics of random Voronoi S-tessellations
Given a locally finite set of points $\{x_1,x_2,\dots\}\subset\mathbb{R}^d$, the Voronoi cell of a point $x_{i}$, denoted by $C(x_{i})$, consists of all the points in $\mathbb{R}^d$ that are closer to ...
- 41
4
votes
1
answer
161
views
Groups (not?) quasi-retracting onto $\mathbb{Z}$ via closest points projection
Inspired by this question we ask:
Suppose that $G$ is an infinite group. Suppose that $X$ is a finite generating set of $G$. Let $\Gamma = \Gamma(G, X)$ be the resulting Cayley graph. Does $\Gamma$ ...
- 28.1k
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
7
votes
0
answers
165
views
Lonely globe trotters
In analogy with the lonely runners conjecture,
imagine "globe trotters" each traveling on a longitudinal great circle at different
(constant, positive) speeds. Each "trotter" ...
- 151k
1
vote
0
answers
69
views
Newton-Kantorovich: theorem geometric
This post is cross-posted from Math StackExchange where I did not receive any response after 5 days. I guess this question might be targeted more towards research level mathematics, so I decided that ...
- 11
0
votes
0
answers
48
views
A question on a quantitative form of Farkas' lemma
Suppose A is an $m \times n$ matrix whose entries are non-negative integers and $\mathbf{b}$ is a vector with rational entries. A version of Farkas lemma implies that if the equation $$A\mathbf{x}=\...
- 6,214
6
votes
2
answers
538
views
Bound on the number of unit vectors with the same pairwise inner products
I want to know the bound on the number of unit vectors $v_i$ in $\mathbb{R}^n$ such that $\langle v_i, v_j\rangle=c$ for all $i\ne j$. I know this can be upper bounded by the number of equiangular ...
- 163
3
votes
1
answer
176
views
A generalization of Barrow's inequality
More than seven years ago. I posted this problem in stackexchange:
Let $ABC$ be a triangle, $P$ be arbitrary point inside of $ABC$. Let $A_1B_1C_1$ be the tangential traingle of $ABC$. Let $A'$, $B'$,...
- 1,776
2
votes
1
answer
246
views
Ramsey type property of the Lipschitz constant
The following problem was proposed by Pietro Majer as an extension of an earlier question of mine on Lipschitz functions.
For $f$ a Lipschitz function on $\mathbb R^n$, we denote by
$$\text{Lip}(f, U) ...
- 6,155
4
votes
1
answer
254
views
On the Lipschitz constant outside the stretch set
Let $f: \mathbb R^n \to \mathbb R^m$ be a Lipschitz map. We define the local Lipschitz constant $Lf$ of $f$ at $x \in \mathbb R^n$ by
$$Lf(x) := \lim_{r \to 0_+} \text{Lip}(f, B_r (x)),$$
where $\text{...
- 6,155
6
votes
1
answer
200
views
Are finitely generated subgroups of $\text{GL}_n(\mathbb{Q})$ virtually special?
This might be a silly question--but are there any examples of finitely generated subgroups of $\text{GL}_n(\mathbb{Q})$ that are known to not be virtually special?
- 103
1
vote
0
answers
99
views
Minimum of the maximum element frequency given the family size and the universe size
[Crossposted at math.stackexchange].
Consider families of sets $\mathcal{F}$ with size $n = |\mathcal{F}|$ and universe $U(\mathcal{F})$ with size $q = |U(\mathcal{F})|$.
I have written and solved ...
- 988
3
votes
0
answers
147
views
Understanding why $\frac{\phi^5}{2}$ solves this 3D optimization problem, where $\phi$ is the golden ratio
I would like to understand the deep meaning of a solution which arises from an optimization problem discussed in a paper of mine since it can be simply stated as $\frac{\phi^5}{2}$, where $\phi := \...
- 1,451