Deprecated; please use a more specific tag.

**1**

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52 views

### A geometric optimization problem: is this non-convex? [closed]

If both the two angles $\angle DAB = \angle CBD = \dfrac{\pi}{6}$ , $C$ is a moving point on the radial $BC$, and $\beta\in\left(0,\dfrac{\pi}{3}\right)$ (see Figure below). Suppose $D$ is always ...

**-3**

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37 views

### A quadrilateral ABCD satisfies the following relationship with respect to any point M in the plane AM^2+ CM^2 = BM^2 + DM^2 [closed]

This quadrilateral could be
1. a parallelogram
2. a rectangle
3. a square
4. a rhombus
5. None of the above.

**-1**

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**0**answers

42 views

### Feature relationship based class separability [closed]

I am a computer science guy, not a mathematician so kindly excuse me if there is any ridiculous error in my problem description.
I have two clusters $C_1$ and $C_2$ in a feature space spanned by $k$ ...

**1**

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**0**answers

124 views

### To what extent can Gielis' so-called “Superformula” be used to improve the efficiency of WIFI antennas? [closed]

In 2003, the Belgian plant biotechnologist Johan Gielis proposed a formula that allows for the description of a wide variety of shapes in 2$d$, 3$d$ and higher dimensions. This is the formula
$$r( ...

**13**

votes

**2**answers

231 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 ...

**6**

votes

**5**answers

514 views

### covering by spherical caps

Consider the unit sphere $\mathbb{S}^d.$ Pick now some $\alpha$ (I am thinking of $\alpha \ll 1,$ but I don't know how germane this is). The question is: how many spherical caps of angular radius ...

**2**

votes

**2**answers

225 views

### The probability distribution for vertex degree in a unit disc graph generated from random points on a plane

Imagine I cover an arbitrarily large plane with randomly placed points at some density $\rho$ s.t. the number of points in any randomly sampled area $A$ (of arbitrary shape and size) is $\approx ...

**14**

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**4**answers

644 views

### Metrics for lines in $\mathbb{R}^3$?

I seek a metric $d(\cdot,\cdot)$ between pairs of (infinite) lines in $\mathbb{R}^3$.
Let $s$ be the minimum distance between a pair of lines $L_1$ and $L_2$.
Ideally I would like these properties:
...

**3**

votes

**1**answer

370 views

### How to partition a graph into N groups with M elements nearest?

Hi.
This problem probably already here but I could not find the right words to find it.
I have a list with 1700 points (geographic coordinates) and a need to separate into 17 groups with 100 ...

**8**

votes

**2**answers

629 views

### Will a ball fired through a focus of an ellipse eventually tend to a horizontal line?

A couple of years ago I came across this phenomenon which appears to be true although I am having difficulty proving it.
F and F' are foci of a billiard table in the shape of an ellipse. A ball is ...

**25**

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**6**answers

2k views

### Is it possible to partition $\mathbb R^3$ into unit circles?

Is it possible to partition $\mathbb R^3$ into unit circles?

**6**

votes

**1**answer

383 views

### What is the shape of the $n$-gon which gives the maximum of a function?

What is the shape of the $n$-gon $P_1P_2\cdots P_n$ which gives the maximum of $A_n$? The quantity $A_n$ is defined by
$$ A_n = \frac{{\sum_{i\lt{j}\le{n}}{\lvert P_i ...

**3**

votes

**0**answers

2k views

### What is the geometric meaning of third derivative of a function at a point? [closed]

What is the geometrical meaning of third derivative of a function at a point?
The question is now open in the sister site ...

**2**

votes

**1**answer

209 views

### What is the shape of the convex $n$ -gon which gives the maximum of a function?

Supposing that the length of every edge of the convex $n$-gon $P_1P_2$$\cdots$$P_n$ is 1, what is the shape of the $n$-gon which gives the maximum of the following function $B_n$? ...

**3**

votes

**1**answer

224 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

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

**14**

votes

**2**answers

354 views

### Repeated random two-steps in $\mathbb{R}^3$: unbounded?

I created a random isometry $T$ of $\mathbb{R}^3$ by generating
a random orthogonal matrix $M$,
uniformly distributed among all such,
and a random displacement $v$, whose coordinates
are drawn from a ...

**3**

votes

**1**answer

166 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 ...

**21**

votes

**5**answers

1k views

### What is a good method to find random points on the n-sphere when n is large?

As part of a more complex algorithm, I need a fast method to find random points of the n-sphere, $S^n$, starting with a RNG (random number generator). A simple way to do this (in low dimensions at ...

**20**

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**2**answers

1k views

### Functions whose gradient-descent paths are geodesics

Let $f(x,y)$ define a surface $S$
in $\mathbb{R}^3$ with a unique local minimum at $b \in S$.
Suppose gradient descent from any start point $a \in S$
follows a geodesic on $S$ from $a$ to $b$.
(Q1.)
...

**2**

votes

**0**answers

81 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. ...

**15**

votes

**5**answers

565 views

### Historical use of figures in geometry

I was surprised to learn from John Stillwell's comment in answer to the
question,
"Can the unsolvability of quintics be seen in the geometry of the icosahedron?",
that
There is not a single ...

**4**

votes

**1**answer

370 views

### Any reference to an algorithm for finding the largest empty circle on a sphere (with great-circle distance)?

Given a set $S$ of 2D points in the plane there are known algorithms for finding the largest empty circle ($LEC$) of the set of points.
The $LEC$ problem is stated in this way: find a $LEC$ whose ...

**5**

votes

**1**answer

137 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

92 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 ...

**8**

votes

**2**answers

783 views

### Non-Kahler Complex manifolds

For a non-Kahler complex manifold $M$, we still have the decomposition of differential forms into differential forms of type $(p,q)$ and we can write $d=\partial+\bar\partial$ and we can define ...

**3**

votes

**2**answers

262 views

### Symmetric convex curve

a) Assume that $\gamma$ is a symmetric convex curve w.r.t. two orthogonal lines (such a curve is the ellipse). Is the following statement true. There exists a $(l,L)$-bi-Lipschitz mapping of the unit ...

**3**

votes

**1**answer

103 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 ...

**8**

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**0**answers

157 views

### Covering of the plane with manholes

Some shapes, such as the disk or the Releaux triangle can be used as manholes,
that is, it is a curve of constant width.
(The width between two parallel tangents to the curve are independent of the ...

**20**

votes

**5**answers

2k views

### Nonconvex manhole covers

One common reason given for the circularity of manhole covers is so they can't fall through the manhole. For convex manhole covers, this property is equivalent to having constant width -- if you have ...

**1**

vote

**1**answer

126 views

### Construction of an integral point set given the set of distances ,its minimal description so as to get a measure of its complexity and its unique identifier.

Given a set of distances between every pair of points of an integral point set P of n points; say
D = {${d_i}$}
Q1. What is the least time complexity
possible/known for recreating the
...

**8**

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**0**answers

248 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 ...

**0**

votes

**0**answers

122 views

### Volume comparison theorems

I am trying to understand the volume comparison theorem.
For instance, if there exists $b$ s.t. the sectional curvature $K$ is bounded above, $K\leq b$, then
$$Vol(B_p(r)) \geq V^b(r),$$
where ...

**2**

votes

**1**answer

58 views

### Covering the annulus of d-cube

Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C+x$ = {$~\lambda c+x|c\in C$} for some $x\in R^d$ is called homothetic copy of $C$. The number
...

**1**

vote

**1**answer

77 views

### A certain lemma concerning a “rearrangement” on a manifold

Let $(M,g)$ be a complete connected compact manifold (with or without boundary). Let $f(r)$ be a decreasing function of $r =$ geodesic distance. If $\Omega \subset M$, then
$$ \int_{\Omega} f(r) dV ...

**1**

vote

**1**answer

144 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 ...

**9**

votes

**1**answer

692 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 ...

**0**

votes

**0**answers

71 views

### Formal definitions for a few lattice packing invariants

I'm a bit hesitant to post this question here because it regards definitions that are likely trivial, and please note that I'm cross-posting this question from Math.Stackexchange (posted nine days ...

**0**

votes

**0**answers

36 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 ...

**1**

vote

**1**answer

86 views

### Riemann isometry vs Euclidean bi-Lipschitz mapping

Assume that $\gamma$ is a rectifiable Jordan curve in the complex plane of length $2\pi$. Then there exists a Riemann isometry $f$ between $\gamma$ and the unit circle $T$. My question is, does this ...

**-3**

votes

**1**answer

2k views

### How to transform a plane into a sphere? [SOLVED] [closed]

Given a 2-dimensional array of MxN heights, how to transform it to a sphere? Every element of this array is just a 3D point (x,y,z) where z represents some height. One has to transform this array into ...

**4**

votes

**1**answer

173 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 ...

**6**

votes

**0**answers

314 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 ...

**1**

vote

**2**answers

515 views

### Trilateration problem

When trying to develop an algorithm for a program, I got with the following problem:
Determine the approximate location of $O$, if you can take finite samples $P_n$ from known locations and always ...

**1**

vote

**1**answer

133 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 ...

**3**

votes

**1**answer

156 views

### convex polyhedron in the unit cube

Let $P$ be a given finite set of points within the $n$-dimensional unit cube. A finite set $Q$ of points within the $n$-dimensional unit cube covers $P$ if $conv(Q) \supseteq P$ where $conv(Q)$ ...

**9**

votes

**1**answer

290 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

**0**answers

100 views

### Blowing up spheres in a face centered cubic (fcc) packing geometry just enough to cover the volume of the lattice

Imagine I have an infinite lattice of spheres packed in a face centered cubic (fcc) lattice geometry which has the basis: $((-1, -1, 0), (1, -1, 0), (0, 1, -1))$. Here, provided that sphere-sphere ...

**4**

votes

**1**answer

277 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

134 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 ...