Deprecated; please use a more specific tag.

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26 views

### How can I efficiently check a point lie in 4 circles on sphere? [migrated]

Given coordinate of 2 points $A,B$ (Cartisian or longitude-latitude coordinate) on sphere of radius $R_1$. Point $O$ is middle of $AB$, 2 points $E$ and $F$ is derivation from $O$ by a distance ...

**2**

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55 views

### Equidistribution of Brillouin zones

Answering the question about Limiting shape for Brillouin zones Victor Kleptsyn proved that $N$th Brillouin zone is very close to a circle of radius $c\sqrt N$ (you can find all necessary definitions ...

**2**

votes

**1**answer

76 views

### How many points are in such set with the same norm-2

Let $L=[a,b]\cap\mathbb{N}$ with $a,b\in\mathbb{N}$, let $D\in\mathbb{N}$, and let $C=L^D$. Then I would like to know how many points are there in $C$ with the same given norm-2 $d$. I.e., I'm looking ...

**3**

votes

**1**answer

96 views

### How to show it is contained in a convex hull?

There are $(d+1)f$ points(denote set of all points as $S$) in $\mathbb{R}^d$, that can be divide into $d+1$ disjoint sets $F_1,...,F_{d+1}$, each set of size $f$. If we have
$$
\mathcal{H}(F_i)\bigcap ...

**11**

votes

**1**answer

265 views

### Limiting shape for Brillouin zones

Is it true that the limiting shape for Brillouin zones (for any lattice) is a circle?
You can find the definition and the step by step construction of Brillouin zones here. This picture is taken from ...

**4**

votes

**1**answer

95 views

### what's the formula of the inradius of a general simplex? [closed]

As the title, I just want to know whether there is a general formula for calculating the inradius of a n-simplex. Thank you!

**1**

vote

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63 views

### Existence of polytope

Does there exist a polytope in dimensional d consisting of $k>d+1$ faces satisfying that every d faces intersect? I tried 3 dimensional cases, and it seems negative. But is it all negative for any ...

**2**

votes

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70 views

### good choice of extension of equivariant map

Let $K$ be a compact Lie group. Let $C_K$ denote the category whose objects are the compact lie groups containing $K$ and whose morphisms are inclusion of the groups. Let $Y$ be a $K-$space such that ...

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votes

**2**answers

74 views

### Maximal-Orthogonal Convex Hull (or Maximal-Rectilinear Convex Hull) [closed]

Edit : Consider giving a reason for down vote.
In my research, I have come across a this paper from the Computational Geometry field and I am not able to understand the concept of ...

**6**

votes

**1**answer

175 views

### Lattice points on the boundary of an ellipse

How many points of the integer lattice ${\mathbb Z}^2$ can an axis-parallel ellipse of radius $r$ contain on its boundary? (that is, we consider ${\mathbb Z}^2$ as lying in ${\mathbb R}^2$). ...

**4**

votes

**1**answer

141 views

### A conjecture for a curve cuts a curve - variant Cayley-Bacharach's theorem

I propose a conjecture variant of Cayley-Bacharach's theorem.
I'm an electrical engineer, I am not a mathematician. I don't know how to prove this result. Could you give a solution or let me know ...

**2**

votes

**1**answer

127 views

### Is it possible to construct an isosceles triangle by using a ruler and without using a pair of compasses?

It is well known that on Euclidean plane one can construct an isosceles triangle on given straight line by using a ruler and a pair of compasses.
Also it is possible to construct straight line ...

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vote

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37 views

### Thomsen Blaschke condition

I am reading a paper (Paper 1: https://ideas.repec.org/p/cwl/cwldpp/76.html, that cites another paper ( Paper 2) for its proof.
Paper 1, page 1, line 10 says : Consider the topological image G of a ...

**9**

votes

**4**answers

554 views

### Is this an instance of any existing convex pentagonal tilings?

Inspired by Wikipedia's article on pentagonal tiling, I made my own attempt.
I believe this belongs to the 4-tile lattice category, because it's composed of pentagons pointing towards 4 different ...

**3**

votes

**1**answer

357 views

### Holomorphic coordinates on a Kähler manifold

Let $(X,J,\omega)$ be a Kähler manifold. Let $\dim_{\mathbb{R}}(X)=2n$ and we also know that $X$ splits as $X=M\times \mathbb{R}$, where $\dim_{\mathbb{M}}=2n-1$. My question is now: does there exists ...

**4**

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**0**answers

277 views

### Averaging lengths and distances

A natural way in which Finsler metrics appear in Riemannian geometry is as averages of Riemannian metrics (e.g., the average of the arc-length elements
$\sqrt{dx^2 + dy^2}$ and $\sqrt{2dx^2 + 3dy^2}$ ...

**3**

votes

**1**answer

249 views

### cover and hide with squares

I am studying two numbers, related to squares, that can characterize a polygon P:
MinCoverNumber = the minimum number of axis-aligned squares required to exactly cover P (the covering squares may ...

**2**

votes

**0**answers

112 views

### Reachability in dynamic random graphs

There has been some advice on how to ask questions. So I reask my question.I have a problem related to graph theory, random graphs or random geometric graphs in special that really confuses me. ...

**5**

votes

**1**answer

186 views

### Matching on sphere to create cycle with chords

Imagine a number of chords of a sphere $S$ which nearly, but not quite, pass through
the center of $S$, in such a way that no pair of chords intersect:
I would like ...

**0**

votes

**0**answers

123 views

### Area on the unit sphere swept out by big circles corresponding to a curve

For a point on the unit sphere, we know the plane perpendicular to the line through the origin and the point cuts the sphere with a big circle. When the point moves along a sphere curve, the ...

**3**

votes

**1**answer

133 views

### Existence of Simple Closed Straightest Geodesics

There are at least three distinct simple closed quasigeodesics on convex polyhedra [Mat. Sb. (N.S.), 1949, 25(67) :2, 275–306 Quasi-geodesic lines on a convex surface Pogorelov].
Is the same true ...

**7**

votes

**0**answers

297 views

### Expanding disks lead to what packing of the plane?

Suppose one sprinkles points uniformly at random on the infinite Euclidean plane,
with some density $\rho$ per unit area.
View the points as disks of radius zero.
Now the radii $r$ of all disks grows ...

**16**

votes

**2**answers

1k views

### Can the unsolvability of quintics be seen in the geometry of the icosahedron?

Q1. Is it possible to somehow "see" the unsolvability of quintic polynomials
in the $A_5$ symmetries of the icosahedron (or dodecahedron)?
Perhaps this is too vague a question.
Q2. Are there ...

**1**

vote

**1**answer

155 views

### Helly's number from biconvex functions

Helly's Theorem states the following. Suppose $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...

**0**

votes

**0**answers

40 views

### Covering the annulus of symmetric convex body

Consider a symmetric convex body $A$ in $R^d$. Now, we draw another object, $A'$, concentric and translated with respect to A and having radius slightly greater than twice to the radius of $A$.
Now ...

**4**

votes

**1**answer

190 views

### Zoll Flat Finsler tori and convex bodies on a starry night

The starry night. The "celestial sphere" is given by set of non-zero vectors in $\mathbb{R}^n$ modulo positive dilations (i.e., $v \equiv w$ if $v = \lambda w$ for some $ \lambda > 0$) and the ...

**7**

votes

**0**answers

354 views

### Rectangology and squareology

I thought that rectangles were simple, and squares even simpler. Until my research has led me to several questions about rectangles and squares, which I can't solve.
I started by posting this ...

**2**

votes

**1**answer

143 views

### Helly's Theorem for Biconvex Sets

Helly's Theorem states the following. Suppose that $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...

**4**

votes

**1**answer

334 views

### Biggest ball included in an intersection of balls

I would like to prove that for any family of balls $\{B(c_i,r_i)\}_i \subset \mathbb{R}^d$ such that $\{c_1, \dots, c_n\} \subset \bigcap_i B(c_i,r_i) $ and $\forall i, r_i \geq 1$, there exists a ...

**10**

votes

**1**answer

317 views

### Needle probing for a convex body

Suppose there is an unknown closed convex body $K$ of
volume vol$(K) = V$ inside the
unit cube $[-\frac{1}{2}, \frac{1}{2}]^d$ in $\mathbb{R}^d$.
You are permitted to probe with a (one-dimensional)
...

**2**

votes

**2**answers

142 views

### A notion of a 'coarse', parametrized dimension of an object, where the parameter determines how finely we can distinguish (say) a very thin rod from a line

I apologize for the clumsy wording of the title-- what I'm looking for is a notion of an integer-valued dimension $d_{\epsilon}$, which we parametrize by a real positive number $\epsilon$, of, say, a ...

**15**

votes

**2**answers

422 views

### Integer lattice points on a hypersphere

Is the following statement true?
For every integer $n\ge2$ and every integer $k\ge0$ there exists a hypersphere in $\mathbb{R}^n$ (circle, sphere etc) containing exactly $k$ integer lattice ...

**0**

votes

**1**answer

309 views

### Relations between automorphisms of field of rational functions and Mobius Transfomation

Proposition: If $F$ is a field, let $F[x]$ be the ring of all polynomials whose coefficients are in $F$. The fraction field of $F[x]$, denoted $F(x)$, is defined to be the ratios $r(x) = f(x)/g(x)$ ...

**2**

votes

**0**answers

34 views

### Diameters of the images of two balls under a function

Let $ \Omega $ be an open and bounded subset of $ \mathbb{R^n} $, and let $ f:\Omega \to \mathbb{R} $ be a continuous function. I'm looking for some (preferably, minimal) conditions on $ f $ under ...

**2**

votes

**1**answer

247 views

### Is there an “accepted” jamming limit for hard spheres placed in the unit cube by random sequential adsorption?

I have a unit cube, and operating in the continuum limit (i.e. not on a lattice), I sequentially place spheres of some radius $r$ inside the cube until a filled volume "jamming limit" ...

**5**

votes

**1**answer

266 views

### Reference request: affine transforms + circle inversion?

This problem cropped up in the context of scale-insensitive methods for generating random variables.
Let $X=R^n \cup \{\infty\}$. Suppose we consider a set of transforms $\cal{T}$ from $X\rightarrow ...

**2**

votes

**0**answers

217 views

### A question on the theorem of Minkowski-Hlawka

The Minkowski-Hlawka theorem (as stated by Gruber in his lovely book Convex and Discrete Geometry) says that if $S$ is a Jordan measurable set in $\mathbb{R}^n$ with volume < 1. Then there is a ...

**1**

vote

**1**answer

115 views

### General and translational Birkhoff lattices. Equational classes.

By lattice I'll mean Birkhoff lattice.
The two classical equational classes of lattices are modular lattices and distributive lattices. The old problem used to be:
Is there an ...

**3**

votes

**1**answer

239 views

### Sums of uniformly random vectors from the $n$-dimensional unit ball

I'm interested in some instances of the following problem.
Let $n \geq 2$, and suppose we draw $k \geq 2$ vectors $v_1, \dots, v_k$ uniformly at random from the $n$-dimensional ball of radius $1$, ...

**2**

votes

**4**answers

9k views

### How to find overlap between two convex hulls,along with the overlap area

I have two boundaries of two planar polygons , say,B1 and B2 of polygons P1 and P2(with m and n points in Boundaries B1 and B2). I want to find out if the Polygons overlap or not.If they overlap,then ...

**4**

votes

**0**answers

347 views

### The probability distribution for the number of pairwise distances $\leq$ some threshold for points uniformly placed in a sphere

If I place place $N$ particles in a sphere of radius $R$, selecting positions across the sphere's volume with uniform probability, what is the exact probability distribution for the number of pairwise ...

**1**

vote

**1**answer

80 views

### Inferring the properties of a visibility blocker tangential to a point-like light source

Imagine there's a point-like particle undergoing radioactive decay at some position $(0,0,0)$ in Euclidean $3$-space. We encapsulate this particle with a spherical detector for the decay products it ...

**1**

vote

**0**answers

102 views

### Boundary surfaces in a 3d Voronoi tessellation with obstacles

Let $x_1,\dots,x_n$ be a set of points in $\mathbb{R}^3$ and let $\mathcal{O}_1 ,\dots, \mathcal{O}_m$ denote a set of polyhedral obstacles. What is the name for the surfaces that describe the ...

**0**

votes

**1**answer

449 views

### Generalization of join of simplicial complexes

The join of two abstract simplicial complexes $K$ and $L$, denoted $K\star L$ is defined as a simplicial complex on the base set $V(K)\dot{\cup} V(L)$ whose simplices are disjoint union of simplices ...

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**0**answers

155 views

### Optimal paintbrush geodesics

Let $S$ be a smooth, closed surface in $\mathbb{R}^3$,
and $\gamma$ a geodesic segment on $S$, i.e., a finite-length piece
of a geodesic.
Define $\gamma(w)$ as all the points of $S$ within
a distance ...

**3**

votes

**2**answers

266 views

### Constructing a special infinite-dimensional vector bundle

Let $M$ and $N$ be finite dimensional smooth manifolds and $p: E \rightarrow N$ a finite rank vector bundle. $M$ can be assumed to be compact if necessary but I would prefer to work without this ...

**5**

votes

**1**answer

379 views

### A question on the Mahler conjecture

In its asymmetric version, the Mahler conjecture states that if $K \subset \mathbb{R^n}$ is a convex body containing the origin as an interior point and
$$
K^* := \{y \in \mathbb{R}^n : \langle y, x ...

**14**

votes

**2**answers

267 views

### Random rings linked into one component?

Let $S$ be a sphere of unit radius.
Let $C_n$ be a collection of unit-radius circles/rings whose centers
are (uniformly distributed)
random points in $S$, and which are oriented (tilted) randomly ...

**3**

votes

**1**answer

161 views

### “Trapping” of discs after random sequential adsorption

Imagine I perform Random Sequential Adsorption (RSA) of discs of some radius $r$ on $[0, 1]^2$, eventually covering the surface to some density $Q \leq 0.543$ with $N$ total discs (where $\approx ...

**30**

votes

**5**answers

863 views

### Can every $\mathbb{Z}^2$ disk be pinball-reached?

Let every point of $\mathbb{Z}^2$ be surrounded by a mirrored disk of radius $r < \frac{1}{2}$,
except leave the origin $(0,0)$ unoccupied by a disk.
Q. Is it the case that every disk can be ...