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# Questions tagged [spherical-geometry]

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### 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 $2$...
1 vote
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### 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 ...
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### Poincaré-Type inequality for vector fields on the sphere

I have a question which is a vector valued "variant" of the classical Poincaré inequality on the sphere. Consider the sphere $S^{n-1}\subset\mathbb{R^{n}}$ and $\nabla_{s}$ the corresponding ...
188 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 ...
84 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 ...
150 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 ...
98 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 ...
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### 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 ...
85 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 ...
30 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 ...
191 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 ...
141 views

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### Are spherical maps with low distortion locally expanding?

$\newcommand{\SO}{\text{SO}(#1)}$ $\newcommand{\Hom}{\text{Hom}(#1)}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\S}{\mathbb{S}}$ The question in a nutshell: Are the "best" spherical maps ...