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4 votes
0 answers
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MGFs of sum of (Rademacher) independent variables and (hyperbolic/spherical) Pythagorean theorem

Consider a set of iid random variables $X_1, X_2, \ldots$ (distribution to-be-specified later). For real numbers $a_1, a_2, \ldots$ (with $\sum_{k} a_k^2 < \infty$) define $$X = a_1 X_1 + a_2 X_2 +...
ccriscitiello's user avatar
7 votes
0 answers
109 views

Defining convex sums locally on the sphere?

$S^1$ and the torus $T^2$ are spaces in which convex combinations don't make sense globally but do locally. Despite their standard representations in $\mathbf{R}^2$ and $\mathbf{R}^3$ respectively not ...
aleph2's user avatar
  • 545
1 vote
0 answers
41 views

Fourier transform relation for spherical convolution

Let $f$ and $g$ be two functions defined over the 2d sphere $\mathbb{S}^2$. The convolution between $f$ and $g$ is defined as a function $f * g$ over the space $SO(3)$ of 3d rotations as $$(f*g)(R) = \...
Goulifet's user avatar
  • 2,226
-2 votes
1 answer
60 views

Inner Products of Elements in Spherical Cap [closed]

I am interested in understanding what is the lowerbound on the inner products of two elements of a sphere. Based on my intuition in dimension 2, I come up with the following conjecture. I appreciate ...
MMH's user avatar
  • 139
5 votes
0 answers
132 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 ...
RandomTensor's user avatar
2 votes
0 answers
105 views

Poincare inequality on the hemisphere

Background: Let $\mathbb{S}^2_+$ be the hemisphere. Then we know that for $f:\mathbb{S}^2_+\to \mathbb{R}$ satisfying (when written in coordinates) $\int_{0}^{2\pi}\int_{0}^{\pi/2}f(r,\theta)\sin(r)dr ...
Student's user avatar
  • 655
10 votes
3 answers
2k views

Is there an absolute geometry that underlies spherical, Euclidean and hyperbolic geometry?

A space form is defined as a complete Riemannian manifold with constant sectional curvature. Fixing the curvature to +1, 0 & -1 and then taking the universal cover by the Killing–Hopf theorem ...
Mozibur Ullah's user avatar
19 votes
1 answer
1k views

What is the largest subset of the sphere such that inner product of any two points in the set is nonnegative

I'm interested in the question of finding the maximum area of $A\subset S^{d-1}$, such that, for all $x,y \in A, \left<x,y\right>\ge 0$. The portion of the sphere lying in the positive orthant ...
RandomTensor's user avatar
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 \...
MMH's user avatar
  • 139
0 votes
0 answers
40 views

Hyperbolic or spherical analogue to the quadrilateral inequality

This is a reference request. Let $x, y, z, w \in \mathbb{R}^n$. Then we have a so-called "quadrilateral inequality": $$ 0 \leq \lVert x-y-z+w \rVert^2 = \lVert x-y\rVert^2 + \lVert z-w \...
Kacper Kurowski's user avatar
2 votes
0 answers
59 views

Minimum area of a region on the sphere in which an octant can be turned through $\text{360}^{\circ}$

Consider an octant $A \subset S^2$ on the sphere, for example the region $(\theta,\phi)\in[0,\pi/2]\times[0,\pi/2]$ in spherical coordinates. What is the subset $B \subseteq S^2$ with smallest ...
Tommy Williams's user avatar
2 votes
0 answers
100 views

Minimum number of points on sphere which cannot be covered by three double caps

What is the minimum number of points on the sphere $S^d \subset \mathbb{R}^{d+1}$ which cannot be covered by $d+1$ double caps? A double cap is defined to be a set $\{x \in S^d: |\langle x,a \rangle| &...
Tommy Williams's user avatar
2 votes
1 answer
194 views

Are these the only first eigenfunctions on a hemisphere?

Let $\mathbb{S}^2_+$ denote the closed upper hemisphere of the unit round sphere in $\mathbb{R}^3$. It is well known that the first positive eigenvalue of the Laplacian on the closed unit sphere is $2$...
Eduardo Longa's user avatar
1 vote
1 answer
77 views

Let $\alpha\in(0,1),d\in\mathbb N^+$ and $X,Y\in\mathbb S^d$ be uniform, what is $\Pr[\lVert X-Y\cdot\sqrt{1-\alpha} \rVert^2\le \alpha]$?

Suppose that $X,Y$ are independent random $d$-dimensional vectors each uniformly distributed on the unit sphere, and let $Z=Y\cdot\sqrt{1-\alpha}$ be a uniformly selected vector on a slightly smaller ...
J J's user avatar
  • 13
3 votes
1 answer
237 views

Dividing a spherical cap into $n$ equal wedges

This is a follow-up of the question Dividing a spherical cap into three equal wedges where the $n=3$ case was shown. Motivation: Optimal ways to cut an orange. In this problem, we have a spherical ...
TheSimpliFire's user avatar
4 votes
1 answer
103 views

Dividing a spherical cap into three equal wedges

Background: Optimal ways to cut an orange. In this problem, we have a spherical orange, and we do not wish to eat its central column which is modelled as a cylinder. Part of the procedure involves an ...
TheSimpliFire's user avatar
3 votes
0 answers
280 views

Minimal overlap required to cover a sphere with caps is greater than expected for many caps

My question is derived from Covering the surface of a sphere with circles with least overlap on Math SE. In the referenced question, the problem of completely covering a sphere with the smallest ...
Oscar Lanzi's user avatar
  • 1,875
7 votes
0 answers
111 views

A spherical geometry claim related to the perspective 3-point problem

I have a simple claim in spherical geometry that has come out of my research into the so-called "perspective 3-point (pose) problem." Here it is: Fix three (distinct) great circles on the ...
Michael Rieck's user avatar
5 votes
1 answer
149 views

Nonexistence of sphere with one conical point [reference request]

It seems to be considered a classical fact that one cannot have a spherical polyhedral/cone-metric on the 2-sphere with precisely one conical point. However, I've never actually seen it proven ...
Tom Sharpe's user avatar
3 votes
0 answers
128 views

Bounds on the expectation of a product of zonal spherical harmonics

Let us consider a $d-1$ dimensional sphere $S^{d-1}$, and for a point $a \in S^{d-1}$ let $Z_{a,k} : S^{d-1} \to \mathbb{R}$ be the zonal spherical harmonic of degree $k$ in the direction $a$, with ...
Jarosław Błasiok's user avatar
2 votes
0 answers
33 views

Decreasing magnitude of spherical centroid (simplex version)

Let $\sigma$ be the uniform measure on $\mathbb{S}^{d-1}\subset \mathbb{R}^d$. For any region $R\subset \mathbb{S}^{d-1}$, let $X_R$ be a random variable which is uniformly distributed across $R$. We ...
Aaron Goldsmith's user avatar
0 votes
2 answers
237 views

Decreasing magnitude of spherical centroid

Let $\sigma$ be the uniform measure on $\mathbb{S}^{d-1}\subset \mathbb{R}^d$. For any region $R\subset \mathbb{S}^{d-1}$, let $X_R$ be a random variable which is uniformly distributed across $R$. We ...
Aaron Goldsmith's user avatar
3 votes
0 answers
197 views

How to calculate the integral of a product of a spherical Hankel function with associated Legendre polynomials

From numerical experiments in Mathematica, I have found the following expression for the integral: $$ \int_{-1}^{1}h_{n}^{(1)}\left(\sqrt{a^{2}+b^{2}+2ab\tau}\right)P_{n}^{m}\left(\frac{a\tau+b}{\sqrt{...
Chris's user avatar
  • 31
0 votes
0 answers
79 views

Integral over $S^{n-1}$ [duplicate]

What is the values of the following integral: $$\int_{w \in S^{n-1}} e^{i\lambda< x,w >} dw.$$ where $\lambda\in\Bbb R, i^2=-1,x\in\Bbb R^n;<,>$ the inner product scalar on $\Bbb R^n$ ...
zoran  Vicovic's user avatar
1 vote
1 answer
268 views

Probability that three vectors of a unit sphere lie on one side of a hyperplane if angle between the vectors are given

As the title says, How to find the probability of vectors a, b, c, on some unit sphere, all lies on same side of some hyperplane passing through the origin. Information present are the angles between ...
hans's user avatar
  • 23
2 votes
1 answer
179 views

The mean of positive points on a unit $n$-sphere $S^n$

My question is similar to The mean of points on a unit n-sphere $S^n$. I have a unit $n$-sphere $S^n$ and a set $P$ of points lying on its surface. I use geodesic distance metric $d(p,q)=\arccos(pq^T)$...
alagris's user avatar
  • 123
3 votes
4 answers
362 views

Prove that $(v^Tx)^2−(u^Tx)^2\leq \sqrt{1−(u^Tv)^2}$ for any unit vectors $u, v, x$

I believe I found a complicated proof by bounding the spectral norm $||uu^T-vv^T||^2_2:=\max_{||x||=1}|(u^Tx)^2-(v^Tx)^2|$. Using the fact that $dist(x,y):=\sin|x-y|$ is a distance function over unit ...
Dan Feldman's user avatar
1 vote
0 answers
49 views

When are ellipsoids in a convex hull of a sequence with prescribed growth rate?

I am currently reading Dudley's 'Uniform Central Limit Theorems' and found two sections which together would have an interesting geometric interpretation for ellipses in Hilbert spaces. I would like ...
Tardis's user avatar
  • 1,243
0 votes
1 answer
196 views

Prove that $(v^Tx)^2-(u^Tx)^2 < 1-(u^Tv)^2$ for any unit vectors $u$, $v$, $x$

Let $u,v,x \in \mathbb R^d$ be three unit vectors. I found a very complicated proof that $(v^Tx)^2-(u^Tx)^2 \leq 1-(u^Tv)^2$. That is $\lVert uu^T-vv^T\rVert^2_2 = 1-(u^Tv)^2$, or that $f(v,x)\leq f(v,...
Dan Feldman's user avatar
2 votes
1 answer
192 views

Geometry in Hilbert spaces / spheres in high dimensions

Let $H$ be a separable Hilbert space of infinite dimension and let $(e_n)_{n \in \mathbb{N}}$ be an orthonormal basis of $H$. For a series $(\alpha_n)_{n \in \mathbb{N}} \subset \mathbb{R^+}$ we are ...
Tardis's user avatar
  • 1,243
3 votes
0 answers
142 views

Matrix equation and spherical harmonics

I have a set of functions expanded in a finite number of spherical harmonics (up to degree $L$), $$ \eta_k^n(\theta,\phi) = \sum_{l=0}^L \sum_{m=-l}^l d_{kl}^{nm} Y_l^m(\theta,\phi) $$ Similar to the ...
user3516849's user avatar
1 vote
1 answer
132 views

The relationship between facets of an inscribed polytope and those facets' shadows

I posted this question thinking that the response would be two or three answers that say "Counterexamples to this are found in every textbook—for example this one and this one and this one." ...
Michael Hardy's user avatar
5 votes
0 answers
209 views

Covering the sphere with an approximately planar grid

Consider a triangulation of a radius $R$ sphere into $n$ triangles. Must $Ω(\sqrt n)$ triangles have $Ω(1)$ relative difference from being an equilateral triangle of area $4πR^2/n$?  ($Ω$ is from ...
Dmytro Taranovsky's user avatar
6 votes
0 answers
426 views

A spherical version of the generalized half-angle formulas

The following is a generalization of the half-angle formulas presented at Nabla - Applications of Trigonometry. Generalization. Let $a$, $b$, $c$, $d$ be the sides of a general convex quadrilateral, $...
Emmanuel José García's user avatar
2 votes
0 answers
156 views

Spherical harmonics, $\frak{sl}_2$, and algebra gradings

Let $S^2$ be the usual $2$-sphere considered as the quotient $S^3/S^1$, and denote by $\operatorname{Pol}(S^2)$ the algebra of polynomial functions on $S^2$. We can decompose $\operatorname{Pol}(S^2)$ ...
Quin Appleby's user avatar
3 votes
0 answers
221 views

Spherical harmonic expansion of a power function

Let $f$ be an even continuous function on the sphere $S^{n-1}$. Find a relation of the spherical harmonic expansion between coefficients of $f^n$ and those of $f$.
user4164's user avatar
  • 109
2 votes
1 answer
85 views

References Request: A paper Tanno's equation

I need a paper which wrote by Tanno where he prove that a equations $f_{ijk} + k(2f_{k}g_{ij} + f_{i}g_{jk} + f_{j}g_{ik})$ can be solved if and only if on sphere. But I can not find it on Internet ...
管山林's user avatar
9 votes
2 answers
1k views

Is there a nice orthogonal basis of spherical harmonics?

Recall that a function is harmonic if its Laplacian is zero. Let $\mathrm{Harm}(n,k)$ denote the vector space of $n$-variate harmonic polynomials that are homogeneous with degree $k$. When working ...
Dustin G. Mixon's user avatar
6 votes
0 answers
249 views

What is the expected value of the volume of a tetrahedron inscribed in the unit sphere?

Four (non-coincident) points on the unit sphere determine a tetrahedron. What is the expected value of the volume of such a tetrahedron--the volume of the sphere itself being $\frac{4 \pi}{3} \approx ...
Paul B. Slater's user avatar
1 vote
0 answers
50 views

How to tile a plane such that moving from one tile to the next in any of the 8 cardinal directions is the same length?

When tiling the euclidean plane with squares (like most board games), moving diagonally to another tile is longer than moving vertically or horizontally. Is there a tiling such that moving in any of ...
YEp d's user avatar
  • 11
0 votes
1 answer
253 views

Explanation of a formula to calculate the zenith distance of sun and moon [closed]

I am studying tidal accelerations and referring to a well known paper by I M Longman : Formulas for computing.." J Geophys Research 64 (12) Dec 1959. At Eq 12 he writes a term "1336.rev"...
davidmorley's user avatar
14 votes
0 answers
399 views

Minimum number of distinct triangles for tesselating the sphere

Consider sequences of tesselations of the sphere. For instance, one such sequence might start with an icosahedron and proceed by subdividing each triangle face into 4 triangles and projecting the new ...
Arthur B's user avatar
  • 1,892
7 votes
0 answers
198 views

"Universal" polynomial of bounded norm on the sphere

Consider the space $V_{d,n}=\mathbb{R}[x_1,\ldots,x_n]_d$ of homogeneous polynomials of degree $d$ in $n$ variables. I am interested in the set of bounded polynomials on the sphere $$B_{d,n}=\{f\in V_{...
Hans's user avatar
  • 2,927
1 vote
1 answer
78 views

Uniqueness of function with range $\mathbb{S}^2$ under a constraint

Assume $g,f\colon A\subset\mathbb{R}^M\rightarrow\mathbb{S}^2$ are two bijective functions defined on the set $A$. Now assume a constraint $C$: $\forall x,y\in A, \exists R\in SO(3)\colon Rf(x)=f(y)\...
solus0684's user avatar
4 votes
1 answer
188 views

Is there a spherical analogue of polar duality for spherical complexes?

Let $P$ be a spherical complex, which essentially means a tiling of a sphere, let us say the $(d-1)$-dimensional sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^d$ to fix notation, where each cell is a ...
Malkoun's user avatar
  • 5,051
13 votes
3 answers
419 views

Maximal distance between $2d+1$ points on the $(d-1)$-sphere

If one arranges $2d$ points on the sphere $\mathbf S^{d-1}\subset\Bbb R^d$ at the vertices of the crosspolytope, then one can achieve a minimal spherical distance of $\pi/2$ between any two points, ...
M. Winter's user avatar
  • 12.8k
3 votes
0 answers
64 views

Minimizing expected mutual distances in spherical regions

Suppose I take the unit sphere in $d$ dimensions, and I take some subset $A$ of the sphere of fixed relative volume $V$. Now from this set $A$ I draw two vectors, uniformly at random, and I look at ...
TMM's user avatar
  • 733
2 votes
0 answers
155 views

Maximizing $\iiint|(x-z)\times(y-z)|d\mu d\mu d\mu$ over probability measures on the unit sphere

This is a follow-up question to the one asked here (the unit circle case). What probability measure(s) maximize the quantity $\iiint_{\mathbb{S}^2}|(x-z)\times(y-z)|d\mu(x)d\mu(y)d\mu(z)$? The ...
Josiah Park's user avatar
  • 3,177
5 votes
1 answer
383 views

Maximizing $\iiint|(x-z)\times(y-z)|d\mu d\mu d\mu$ over probability measures on the unit circle

What probability measure(s) maximize the quantity $\iiint_{\mathbb{S}^1}|(x-z)\times(y-z)|d\mu(x)d\mu(y)d\mu(z)$? The answer appears to be uniform measure, since informally it appears better to have ...
Josiah Park's user avatar
  • 3,177
5 votes
2 answers
234 views

Maximum number of half great circles of length $\pi$ can be drawn on a sphere without any intersection

It is well-known that any two great circles intersects on a sphere. In fact, there are infinitely many half great circles can be drawn on a sphere with a common intersection. Intuitively, it seems to ...
James Cheung's user avatar
  • 1,865