4
votes
2answers
95 views

The relationship between two definitions of star-shaped domain

There are two definitions of star-shaped domain. One is given in wikipedia as follows. Def1: A set $S$ in the Euclidean space $\mathbb{R}^n$ is called a star domain (or star-convex set, star-shaped ...
4
votes
1answer
74 views

Division of plane into equal area regions

We divide a plane ($\mathbb{R}^2$) into infinite number of regions each of area equal $1$. We can use only (one-dimensional) curves which may meet at points. Fix a point $p$ on a plane and consider ...
0
votes
0answers
29 views

Geodesics of a cone

To find Geodesics on a cone I used the cylindial coordinates $x=rcos\theta$ $y=rsin\theta$ $z=z$ Is this parameterization correct.How can I know how to parameterize? Then arc length $ds^2=r^2 ...
3
votes
2answers
63 views

What is the most elementary but still correct according to the most rigorous standard proof of the isoperimetric inequality?

Can you write the most elementary proof of the isoperimetric inequality (but still correct according to the most rigorous standard )? $$l^2> 4πA$$
11
votes
1answer
345 views

A variation of the isoperimetric problem in the plane

The isoperimetric problem in the plane: « The classical isoperimetric problem dates back to antiquity. The problem can be stated as follows: Among all closed curves in the plane of fixed ...
3
votes
1answer
170 views

Finding the shortest path length on a curved surface(hyperboloid)

I wish to find the minimum path length between two points $P_1(\sqrt2,0,-1)$ and $P_2(0,\sqrt2,1)$ on a hyperbolic surface $S =\{(x,y,z)\in R^3\ |\ x^2+y^2-z^2=1\}$ I faintly recall studying ...
1
vote
1answer
43 views

Measure of surface quadrature

In the article you can find at http://www.cc.gatech.edu/~turk/my_papers/schange.pdf, precisely at page 2 of the .pdf, there is a functional E which is said to be a measure of the aggregate squared ...
6
votes
3answers
176 views

Beach Path math question

Anyone who has walked on the beach knows that walking speed is dependent upon how far away from the ocean one walks. If you walk on the wet sand you can walk much more quickly than if you walked on ...
5
votes
1answer
280 views

Geometric proof of the geodesics of a sphere?

I have seen the standard variational proof that great circles are the geodesics on the $2$-sphere. Do you know a purely geometric proof of this fact, not involving calculus of variations or ...
1
vote
1answer
84 views

Maximium of area/perimeter^2 of a function

Let $a,b$ be two reals. Does there exists a constant $C$ such that for all functions $f:[a,b]\to\mathbb{R}$, continuous on $[a,b]$ and differentiable on $(a,b)$ with $f(a)=f(b)=0$, \begin{equation} ...
0
votes
1answer
80 views

Lines through the origin and Euler-Lagrange

What form should the minimal-length curve to $\int \sqrt{dr^2 +r^2d\theta^2\over{1-r^2}}$ take? I think I can use the Euler-Lagrange equations. So write the integral as $\int\sqrt{({dr\over ...
2
votes
1answer
85 views

Infimum length of curves

Let the unit disc $\{(x,y): r^2=x^2+y^2<1\}\subset\mathbb R^2$ be equipped with the Riemannian metric $dx^2 +dy^2\over 1-(x^2+y^2)$. Why does it follow that the shortest/infimum length of curves ...
5
votes
1answer
641 views

Shortest path on hyperboloid

On the sphere $S^2$, the shortest path between two points is the great circle path. How about $H^2$, the hyperboloid $x^2+y^2-z^2=-1, z\ge 1$, with the Euclidean distance? Is there a formula for the ...
12
votes
8answers
5k 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 ...