Tagged Questions
9
votes
4answers
217 views
What will be a circle look like considering this distance function?
I am working on some exercises in the book Geometry: A Metric Approach with Models by R.S. Millman. He defines the following map: $$d_S(P,Q):\mathbb R^2\times\mathbb R^2\to\mathbb R\\\ ...
3
votes
3answers
132 views
Topological boundary vs geometric boundary
Let $M_1=B((0,0),1)=\{(x,y) \mid x^2+y^2<1\}$
$M_2=\{(x,y) \mid x^2+y^2\le1\}$
What are the interior of $M_1$ and $M_2$ ?
And What are the boundary of $M_1$ and $M_2$ ?
How to find them? ...
1
vote
2answers
45 views
Parametrization of $n$-spheres
This comes from Guillemin and Pollack's book Differential Topology. The book claims that one cannot parametrize a unit circle by a single map. I thought we could (by a single angle $\theta$).
I ...
0
votes
0answers
61 views
Geometrical Inequality
Let $ABCD$ be a quadrilateral on the unit circle, and the diagonals
$AC$ and $BD$ intersects at $E$. If the shortest height of the
triangle $ACD$ equals the radius of the incircle of the triangle ...
1
vote
1answer
63 views
Question on the perimeter of any quadrilateral
Is it true that the perimeter of any convex quadrilateral inside a unit circle
is no more than $4\sqrt{2}$?
1
vote
1answer
82 views
Does there exist such a pentagon that can be covered by a circle?
Does there exist a pentagon in which every two nonadjacent vertices
is connected by a diagonal and the minimal height of every triangle formed by the
sides and diagonals of the pentagon whose two ...
0
votes
1answer
15 views
An approximation of lengths dealing with elements of cubes. Show $|x-y| \approx \ell(Q) + |x - c_Q|$.
So the problem I have at hand should be rather elementary, but I can't figure out. It's in a proof I'm trying to understand completely. Here's what I need to do exactly:
Show that $|x-y| \approx ...
4
votes
2answers
67 views
Showing the function $f(x,y)$ is one by one
Yesterday, while teaching geometry, I was faced to a problem saying that the function below is an distance function: $$d(P,Q)=\Big|\ln\frac{\frac{x_1-c+r}{y_1}}{\frac{x_2-c+r}{y_2}}\Big|$$ where in ...
0
votes
3answers
109 views
Intersection of chord with circle knowing the length and a point
Let's take a circle with radius R, and center in O (0, 0). We take on this circle a point A with coordinates xA and yA.
We know that point A is one of the endings of a chord with length l.
Which is ...
2
votes
1answer
64 views
Closed Convex sets of $\mathbb{R^2}$
Can some one please list the closed convex sets in $\mathbb{R^2}$ up to homeomorphism. How many of them are compact
1
vote
2answers
42 views
Finding the boundary of a set given by inequalities
Let $f_1, \ldots f_m:\mathbb R^n \to \mathbb R$ be continuous, and suppose the set $E$ is given by $$E=\{ x \in \mathbb R^n: \forall i,f_i(x) \ge0 \}$$
Is it true that $$ \partial E=\{x \in \mathbb ...
2
votes
1answer
59 views
Equilateral triangle in $l_p^2$
I was trying to calculate the side of an equilateral triangle with the vertices on the unit sphere of $l_p^2$, when $1<p<\infty$. When I say "equilateral", I mean with respect to the distance in ...
5
votes
2answers
83 views
a set of diameter $d$ is contained in a ball of diameter $d$?
Suppose $S$ is a connected open subset of $\mathbb{R}^n$, and $d$ is the diameter of $S$. Is $S$ contained in some ball of radius $d/2$?
1
vote
1answer
32 views
Maximizing the Volume of Body under a Function
Given is the function $y = 1/2 (4-x)\sqrt{x} $
One has to calculate a) the volume of revolution between function and x-axis, restricted by the function's zeros. b) What is the biggest volume possible ...
7
votes
1answer
77 views
Geometric inequality with a triangle
The positive real numbers $x,y,z$ are the side lengths of a triangle iff $$x^2 + y^2 + z^2 < 2\sqrt{x^2y^2 + y^2z^2 + z^2x^2}$$
21
votes
2answers
321 views
New twist on a Putnam problem
A recent Putnam problem:
Let $f$ be a real-valued function on the plane such that for every square $\square ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f$ is identically ...
5
votes
1answer
136 views
Interior Sphere Condition
Let $\Omega\subset\mathbb{R}^N$ be a bounded open set. We say that $\Omega$ satisfies the interior sphere condition (ISC), if for all $y\in\partial\Omega$ there is $x\in\Omega$ and a open ball ...
3
votes
2answers
103 views
Does $d(x+u, y + v) \le d(x, y) + d(u,v)$ holds for every metric?
The title said it, I want to prove that
$$
d(x+u, y + v) \le d(x, y) + d(u,v)
$$
for every metric $d$. If the metric is induced by a norm, i.e. $d(x,y) := ||x-y||$, then this is easy.
\begin{align*}
...
4
votes
0answers
60 views
Spectrum of laplacian in a parallelogram
Is the spectrum of the laplacian on an arbitrary parallelogram with dirichlet boundary conditions known?
19
votes
0answers
351 views
Ambiguous Curve: can you follow the bicycle?
Let $\alpha:[0,1]\to \mathbb R^2$ be a smooth closed curve parameterized by the arc length. We will think of $\alpha$ like a back track of the wheel of a bicycle. If we suppose that the distance ...
4
votes
0answers
73 views
How many points does one need for an epsilon-net
Does anyone know, how many points does one need to have an $\varepsilon$-net on a unit sphere sitting in the three-dimensional Euclidean space? Thanks!
0
votes
2answers
86 views
Show that the closure of an open ball is exactly $B(0,R) := \{ x \in X : ||x|| \le R\}$
Let $X$ be a normed space and
$$
B(0,R) := \{ x \in X : ||x|| < R \} \qquad \overline{B}(0,R) := \{ x \in X : ||x|| \le R \}.
$$
I want to show that $\overline{B(0,R)} = \overline{B}(0,R)$. I ...
1
vote
1answer
58 views
Show that a certain point lies outside a ball, might be simple but i am stuck…
Consider the ball
$$
B(0, R) := \{ x | ||x|| \le R \}
$$
and consider a point $x$ outside of the ball, that is $||x|| > R$. Now i construct another ball of radius $\frac{1}{2}(||x|| - R)$ around ...
1
vote
0answers
34 views
A certain type of points in the plane
In the plane a point $O$ and a sequence of points $ P_1 , P_2 , P_3 , ... $ are given. The distances $ OP_1 , OP_2 , OP_3 , ...$ are $ r_1 , r_2 , r_3 , ....$ , where $ r_1 ≤ r_2 ≤ r_3 ≤ $ ... . ...
6
votes
3answers
214 views
$\pi$, Dedekind cuts, trigonometric functions, area of a circle
(I should say at the outset that this question is broad, and may need splitting up. Although I ask several questions, I present them as one because they are not independent of one another, and I am ...
2
votes
2answers
201 views
Pattern matching circle, square or triangle
I have a set of x, y co-ordinates that are actually taken from hand drawings of a circle, square or a triangle. Using the set of points, I need to mathematically find if the points approximately fit a ...
3
votes
3answers
182 views
Level curves on ellipsoid
Let $a,b,c>0$ with $a\leq b\leq c$. Let $E$ be the ellipsoid determined by $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$$
Is there a function $f:E\rightarrow \mathbb{R}$ such ...
0
votes
2answers
41 views
Is there any significant Geometric/analytic property of the Pearson coefficent that could be applied to statistics?
Is there many significant Geometric/analytic property that the Pearson coefficent has which could be applied to statistics?
It seems very interesting to me.
Thanks in advance.
4
votes
0answers
140 views
“Green Globs” question
When I was in high school geometry, we had a fun little game on the computer called Green Globs (the website for the software is http://www.greenglobs.net/index.html). A number of targets (globs) are ...
1
vote
2answers
154 views
normal to the surface of revolution
Please, help me with my homework...
Prove that the normal to the surface of revolution $z = f(\sqrt{x^2+y^2})$
intersect the axis of rotation.
first, I need to find $z_x$, $z_y$
$z_x$ = ...
10
votes
2answers
294 views
A property of circles
Consider a set of circles on a plane that don't overlap each other and any of them touches at least 6 other circles. Prove that this set has infinite number of circles.
Well, there seems to be ...
0
votes
1answer
79 views
Function for the upper left part of a circle
What is the function corresponding to the upper left quarter of a circle ? Where $x$ goes from 0 to $x_\text{max}$, and $y=f(x)$ goes from $y_\text{min}$ to $y_\text{max}$.
2
votes
1answer
103 views
Ratio of geodesic segments on the sphere
Let $\mathbb{S}^2$ be the unit sphere. Let $0<\lambda<1$ be fixed. What is the smallest number $0<\mu<1$ (depending on $\lambda$) such that for any three points $A,B,C\in \mathbb{S}^2$, ...
2
votes
1answer
70 views
Constructable Trigonometric Inverses
By doing some right triangle gymnastics, we can derive things like
$\cos(\arctan x) = \frac{1}{\sqrt{1+x^2}}$, for $x>0$
$\cos(\arcsin x) = \sqrt{1-x^2}$
$\tan(\arcsin x) = ...
4
votes
2answers
227 views
Space-filling curve with distance locality
Is there a space-filling curve $C$ that has the property that, if $C$ passes through $p_1=(x_1,y_1)$
at a distance $d_1$ along the curve, and through $p_2$ at $d_2$, then if
$|p_1 - p_2| \le a$, then ...
10
votes
1answer
305 views
Surface area of a convex set less than that of its enclosing sphere?
Is the boundary measure ("surface area") of a convex set in $\mathbb{R}^n$ less than the boundary measure of it's enclosing hypersphere (smallest hypersphere that contains the set)?
In 2D I've found ...
8
votes
1answer
187 views
Can the Banach-Tarski Paradox be extended to an arbitrary number of duplications?
In this question, I recently asked if there were free subgroups of rank 3 or higher of the group of rotations in $\mathbb{R}^3$. From the answers, it follows that any free subgroup of rank 2 admits ...
3
votes
1answer
71 views
Intersection of Sphere and Line in $\mathbb{R}^n$?
This seems to me as a very simple and basic question, though I'm having trouble with it.
The Problem
Given a sphere $K\in\mathbb{R}^n$ with radius $r\in\mathbb{R}$ and center ...
7
votes
2answers
252 views
Are rotations of $(0,1)$ by $n \arccos(\frac{1}{3})$ dense in the unit circle?
Under which conditions will successive rotations of $(0,1)$ by an angle $\theta$ guarantee that given $\delta > 0$ and some point $p$ on the unit circle, there exists some $n$ such that rotating ...
157
votes
13answers
11k views
2
votes
1answer
321 views
A system of geometric equations
I have a question about solving a system of geometric equations. I really hope someone here can help me, it's been several months since I try to solve the problem but without success. As I am not an ...
12
votes
7answers
2k views
Why Circle encloses largest Area?
In this wikipedia, article http://en.wikipedia.org/wiki/Circle#Area_enclosed its stated that the circle is the closed curve which has the maximum area for a given arc length. First, of all i would ...
