# Questions tagged [geometry]

Deprecated; please do NOT use this tag, use a more specific tag.

375
questions

**0**

votes

**0**answers

55 views

### Axioms of betweenness

In Axiomatic Geometry, groups of axioms for points and lines are introduced. The second of these groups are called the axioms of betweenness. Given three points $A,B,C$ we postulate a relation $A*B*C$,...

**2**

votes

**1**answer

218 views

### How can the same polytope have three different volumes? [closed]

I'm quite new to geometry and I came across the idea that the same convex polytope can have at least three different volumes.
Consider the permutohedron, formed by the convex hull of the n! points ...

**1**

vote

**0**answers

79 views

### Minimum volume of intersection between two high-dim $\ell^1$-balls

Let $B_1$ and $B_2$ be two balls with the same radius, in $\mathbb R^n$ with the $\ell^1$ norm. The distance between the centers of $B_1$ and $B_2$ is $d(B_1, B_2)$. Is there any deterministic method ...

**1**

vote

**1**answer

116 views

### Identity involving dot product of solid angle and gradient [closed]

How to prove following for $n\geq0$ ?
$$\int_{4\pi}d\vec{\Omega}(\vec{\Omega}\cdot\vec{\nabla})^{2n}f(\vec{r})=\frac{4\pi}{2n+1}\nabla^{2n}f(\vec{r})$$
Where, at any point $\vec{r}$, the $\vec{\Omega}$...

**0**

votes

**1**answer

577 views

### State-of-the-art geometry book? [closed]

For my best friend's birthday, I am looking for a geometry book. He's currently doing his math PhD and is really fond of geometry, especially hyperbolic or higher-dimensional ones, also interested in (...

**3**

votes

**1**answer

217 views

### Does there exist concept of angles and sides in Non-archimedian field geometry? [closed]

Question about Non-Euclidean Geometry, particularly about Non-archimedian Geometry:
I am studying and trying to understand about Non-Euclidean Geometry for my future purpose.
An example of Non-...

**5**

votes

**3**answers

366 views

### Is there a dense subset on closed Jordan curve $C$ which its points make intersections under certain rotations?

Is it true that for any given closed Jordan curve of $C \subset \mathbb{R}^2$ there is a dense subset $A$ such that for every point $p\in A$ we have the following property:
If we rotate $C$ around $p$...

**2**

votes

**0**answers

116 views

### Functional inequality under mean curvature flow

Let $\Sigma$ be a hypersurface in $\mathbb R^n$ and $\Sigma_t$ be a variation of $\Sigma$ under the mean curvature flow under an extra condition that ${\rm vol}_{n-1}(\Sigma)={\rm vol}_{n-1}(\Sigma_t)$...

**1**

vote

**1**answer

123 views

### Need help maximizing distances to nearest neighbor in a cylinder

I have a cylinder and I want to maximize the number of points in the cylinder such that the distances to the nearest neighbors are maximally spaced. How do I find out how many points I can have so ...

**3**

votes

**1**answer

199 views

### geometrically infinite ends of hyperbolic 3 manifolds

Let $M$ be a hyperbolic 3-manifold with finitely generated fundamental group. Assume $E$ is a geometrically infinite end (not of geometrically finite type, i.e. the convex core can not be separated of ...

**0**

votes

**0**answers

49 views

### Continuous Functions On Grassmannans under containment restrictions

Let $V$ be a vector space. Suppose that for a $x\in V$, we are given some subspace of dimension no more than d (e.g., the kernel of some operator defined on V, which varies smoothly with x), call it $\...

**2**

votes

**1**answer

117 views

### Exist $A_1, A_2,\cdots , A_n$ be $n$ points on a sphere, satisfy if $i-j=l-m$ then $A_iA_j=A_lA_m$

Conjecture: Let $n\geq4$. Is there a set of $n$ non-planar point $A_1, A_2,\cdots , A_n$ be $n$ on a sphere (three-dimensional space) satisfying the conditions: if $i-j=l-m$ then $d(A_i, A_j) = d(A_l, ...

**12**

votes

**1**answer

457 views

### Find structure geometry of $A_1, A_2,…,A_n$ such that $\prod_{i<j} A_iA_j$ is maximum

In any triangle we have the well-known inequality:
$$\sin{A}\sin{B}\sin{C} \le \frac{3\sqrt{3}}{8} (1)$$
Signification of inequality (1): Let three points $A, B, C$ lie on a circle then $AB.BC.CA$ ...

**1**

vote

**1**answer

435 views

### Intersection between a line and a cube in $D$ dimensions

Say we have a space of dimension $D$. Say we have a $D$-cube of side $l$ centered at the origin and inside it we have a point $P\in \mathbb{R}^D$ and a collection of $D-1$ angles $\phi_1, \phi_2, \...

**2**

votes

**0**answers

91 views

### Determine sub-polygon from line segments with known member connectivity

Test Polygon:
Consider the following polygon as attached. Let the known parameter be as follows:
•Member to member connectivity, i.e. it is known that A – B, X – E, F – B, etc. for all the members.
•...

**3**

votes

**0**answers

280 views

### Does a rectangle exist on any Jordan curve?

Let $C$ be a Jordan curve in $\mathbb{R}^2$. Does there exist points $P,Q,R,S$ on $C$ such that quadrangle $PQRS$ is a non-degenerate rectangle?

**10**

votes

**2**answers

502 views

### Minimum separation among $m$ random points on an $n$-dimensional unit sphere

Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be
$$
\rho = \min_{i,j\in{...

**6**

votes

**0**answers

196 views

### An inequality in cyclic polygon and tangential polygon

I proposed my conjecture, it is strengthened version of the Erdős–Mordell inequality as following:
Let $A_1A_2.....A_n$ be a cyclic polygon and $B_1B_2....B_n$ be the its tangential polygon. Let P be ...

**4**

votes

**1**answer

375 views

### A generalization of Erdős–Mordell inequality [closed]

I proposed my conjecture generalization of Erdős–Mordell inequality as following:
Let $A_1A_2....A_n$ be a polygon in a plane, $P$ be the point in $A_1A_2....A_n$. Let $d_i$ be the distances from $P$ ...

**2**

votes

**1**answer

138 views

### A generalization of the Tucker circle theorem and the Thomsen theorem associated with a conic

I gave a generalization of the Tucker circle theorem and the Thomsen theorem at here. Now, I give a more generalization of these theorems as following:
Problem: Let $A_1A_2A_3A_4A_5A_6$ be a hexagon, ...

**2**

votes

**0**answers

116 views

### An elementary question about metrics on the real plane [closed]

Given the metric $d_p$ on the real plane,
i.e.
$$ d_p(x,y) = d_p((x_1, y_1), (x_2, y_2)) = [|x_1 - x_2|^p+ |y_1 - y_2|^p]^{1/p} $$
for which values of $p$ ($\geq 1$) is it true that the following ...

**1**

vote

**2**answers

159 views

### Descartes' theorem and Circle Packing [closed]

There's something I am missing comparing Descartes' theorem for three isometric circles here and this wiki post on circle packing of 3 circles here.
From my calculation:
$$
r_{ext} = \frac{r_{int}}{{...

**-1**

votes

**1**answer

171 views

### When is the orbit space of a manifold still a manifold of the same dimension?

$\mathbf{Question}$. Let us assume that $M^n$ is a topological manifold of dimension n, with a group action $\Gamma$, which acts discontinuously and freely on $M^n$. Is the orbit space $M^n / \Gamma$ ...

**2**

votes

**0**answers

217 views

### A chain of six circles associated with six points on a circle (in Mobius plane) [closed]

I found a conjecture: A chain of six circles associated with six points on a circle (in Mobius plane).
This is a generalization of the last my previous question in Three chains of six circles. (...

**3**

votes

**0**answers

213 views

### A conjecture on six planes [closed]

When I read Cox's Theorem, and Clifford's Circle Theorem and Miquel six circles theorem, I found the conjecture as folowing. And I checked the conjecture by the Geogebra sofware, the conjecture is ...

**4**

votes

**1**answer

395 views

### Relation of some Euclidean geometry theorems and more conjecture generalizations

In this topic I want to share relation of the Pythagorean theorem, the Stewart theorem and the British Flag theorem, the Apollonius' theorem and the Feuerbach-Luchterhand. Since that I posed two ...

**12**

votes

**0**answers

2k views

### Dao's theorem on six circumcenters associated with a cyclic hexagon

This questions from Ngo Quang Duong's paper
In 2013, O. T. Dao published without proof a theorem with title Another seven circles theorem in Cut the Knot, a free site for popular expositionsof many ...

**5**

votes

**1**answer

354 views

### Sixteen points circle - A conjecture on Möbius plane

The conjecture refer the reader about the Bundle's theorem configuration. (This conjecture from a note)
Consider the Bundle theorem configuration :
Points $A_1, A_2, A_3, A_4$ lie on a circle,
...

**3**

votes

**0**answers

139 views

### Isometric embedding for manifolds with conical singularities?

Motivation:
In the 2+1 dimensional gravity theory, solutions of Einstein equation are locally with constant curvature except at the locus of sources. In this paper the authors investigate solutions ...

**3**

votes

**1**answer

108 views

### The volume of a region arising from planar linkages

Let $x_0,\dots,x_n$ be a collection of variable points in $\mathbb{R}^2$ and let $c>0$ be a fixed constant. Is there any way I could compute an upper bound of the volume of the region in $\mathbb{...

**10**

votes

**1**answer

258 views

### Optimization of points on a plane

Suppose we have $n$ points on a plane. Let $D$ be the sum of the squares of all the pairwise distances between the points. Let $A$ be the area of the convex hull. What is the minimum possible value of ...

**3**

votes

**0**answers

71 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 ...

**12**

votes

**1**answer

577 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 ...

**1**

vote

**0**answers

80 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

**0**answers

79 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 $...

**-2**

votes

**2**answers

345 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 Maximal-...

**7**

votes

**1**answer

812 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

192 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 ...

**1**

vote

**0**answers

66 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 2-...

**4**

votes

**1**answer

609 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**

votes

**0**answers

351 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 $\sqrt{2dx^2 + 3dy^2}$ ...

**2**

votes

**0**answers

140 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. ...

**0**

votes

**0**answers

143 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 ...

**1**

vote

**1**answer

163 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 \...

**4**

votes

**1**answer

217 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

407 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

157 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 ...

**5**

votes

**1**answer

407 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 ...

**2**

votes

**2**answers

150 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 ...

**16**

votes

**2**answers

874 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 points ...