0
votes
0answers
17 views

Finsler Metric from page 2 of the book by Chern and Shen.

Physicist here not a mathematician. I am trying to understand the notation for the Finsler metric in Chern and Shen's book. The equation is $$\textbf{g}_y(u,v):=\frac{1}{2} ...
2
votes
1answer
103 views

Length minimizing curves are geodesic segments

I have a metric space $(X,d)$, a geodesic arc is defined to be a continuous function $\gamma : [a,b] \rightarrow X$, $a < b$, which is (globally) distance preserving and geodesic segments are ...
1
vote
0answers
77 views

How can I define a measure of similarity between two line segments in $\mathbb{R}^2$?

I have two segments in $\mathbb{R}^2$ and I would like to define a measure of similarity between the two segments. My idea is that I can apply: a scale transformation $s$ in order to equate the ...
0
votes
2answers
60 views

Shapes bounded only by lines

What is a term for the set of geometric shapes in the plane, that are bounded by one or more continuous closed curves? This set contains simply-connected polygons and circles but also polygons with ...
3
votes
0answers
23 views

Is there a name for this partial order between metrics?

Suppose we have a set $X$ and two metrics $d_1,d_2$ on it (which may or may not attain $\infty$). Assume furthermore that $d_1,d_2$ have the same metric components (where a metric comoponent is a ...
2
votes
1answer
154 views

Hausdorff distance and intersection

The question is related to the Hausdorff distance between sets, $d_H(S,S')$, which is the greatest of all the distances from a point in one set to the closest point in the other set. Suppose there ...
1
vote
1answer
31 views

Convergence of circles in Hausdorff distance

Every triple of real numbers $(x,y,r)$, where $r>0$, represents a circle with center $(x,y)$ and radius $r$. Suppose we have a infinite sequence of such triples $(x_i,y_i,r_i)$, that converges to ...
7
votes
0answers
84 views

Knight's metric: ellipse and parabola.

Knight's metric is a metric on $\mathbb{Z}^2$ as the minimum number of moves a chess knight would take to travel from $x$ to $y\in\mathbb{Z}^2$. What does a parabola (or an ellipse) became with this ...
0
votes
1answer
98 views

Are Euclidean distances a monotone function of inner products?

Does the sum of all pairs of inner-products of k vectors (real) have to decrease if the sums of Euclidean distances between all pairs of $k$ vectors happens to decrease? Similarly-if decrease is ...
2
votes
1answer
64 views

Where does the power $2$ come from in the Pythgorean theorem?

So $$a^2+ b ^2 =c^2$$ in a right triangle, but where does the power $2$ come from? I know we can use different metrics in the Euclidean space. If we use the $p$-metrics, where $p$ is in place of ...
0
votes
0answers
51 views

Pyramid Question

Let us suppose we have a right pyramid,that is a pyramid with all edges equal to each other and a height which goes to the center of the geometric shape that serves as it's base. We than also know ...
0
votes
0answers
61 views

Hadamard space: property of the Busemann function

I have a question about a property of Busemann functions on Hadamard spaces. Let $X$ be a complete CAT($0$) space. If $r:[0, \infty) \to X$ is a geodesic ray, and $x\in X$ the Busemann function is ...
0
votes
0answers
28 views

Is the diameter of the geodesic closure of a set B bounded purely in therm of the diameter of B?

By a "geodesic metric space" I mean a metric space s.t., for every points x y in X, there is (at least) a geodesic betwenn x and y. A typical example is a metric graph (a graph whose edges have ...
0
votes
1answer
39 views

Squares with supremum metric?

Open balls with the sup metric are actually squares? Why? The sup metric is $$d_s(x,y) = \sup \{ |x_i - y_i | \}$$ and the open ball with sup metric is $$B_r(x) = \{x : d_s(x,y) < r \}$$ If it ...
1
vote
1answer
128 views

Finding the “differentness” of two point clouds

I would like to reduce the "differentness" of two point clouds $X$ and $Y$ to a single comparable value $\lambda$, which would ideally be $0$ when $X$ and $Y$ are identical upto isometry (rotation, ...
0
votes
1answer
60 views

Distance Metric in 4 dimensions $\Bbb R^3\times SO(2)$

The euclidean distance metric, $\sqrt{dx^2+dy^2+dz^2}$, shows the shortest distance between two points in $\Bbb R^3$. What would be the distance metric to show the shortest distance between two ...
1
vote
1answer
43 views

Only constant curves rectifiable

I would need a hint on the following standard exercise. Let $0 < a < 1$ and $(X,d)$ a metric space. Let $d^a$ be the corresponding snowflake metric $d^a(x,y) = (d(x,y))^a$. Show that the only ...
2
votes
3answers
204 views

Find the farthest points in d-dimensional space

We have $n$ points with $d$ coordinates each and we want to find two of them for which distance between them is the biggest, in Manhattan metric. The obvious algorithm has complexity $O(n^2 \cdot d)$ ...
1
vote
2answers
142 views

How well can we embed graphs with shortest path metric into $\mathbb{R}^2$ with Euclidean metric?

If we take the integer lattice in $\mathbb{R}^2$ and make edges from $(m,n)$ to $(m+1,n)$ and $(m,n+1)$, you get your typical city block street layout, and if we put the shortest path metric on the ...
3
votes
2answers
132 views

Distance metric for (infinite) lines in 3D

I would like a metric $d(\;)$ between pairs of (infinite) lines in $\mathbb{R}^3$, with these properties: If two lines $L_1$ and $L_2$ are parallel and separated by distance $x$, then $d(L_1,L_2) = ...
3
votes
5answers
102 views

Minkowski sum of two disks

An open disk with radius $r$ centered at $\mathbf{p}$ is $D(\mathbf{p}, r)=\{\mathbf{q} \mid d(\mathbf p, \mathbf q) < r\}$, and the Minkowski sum of two sets $A$ and $B$ is $A \oplus B=\{\mathbf p ...
2
votes
2answers
53 views

Isometric embedding of a finite set into $\mathbb{R}^n$

Consider a finite set $A$ with a distance $d$. It is easy to check whether $(A,d)$ is isometric to a finite subset of $\mathbb{R}$. Is there a known algorithm to check whether $(A,d)$ is isometric to ...
2
votes
1answer
66 views

Group acting by isometries on a length space

I am reading the book A course in metric geometry by Burago, Burago and Ivanov. I have some difficulties with an exercise 3.4.6 on page 78. The exercise is the following: Let a group $G$ act by ...
1
vote
0answers
119 views

a problem on metric spaces

I am reading the book by Burago and Ivanov "A course in metric geometry". I tried to do some problems but have some difficulties. For example, page 66 exercise 3.1.26: Let $(X, d)$ be a metric space ...
5
votes
2answers
127 views

Why generalize the Euclidean metric?

It is well known that the Euclidean metric can be generalized to $\Bbb R^n$ by $\sqrt{(x_1-x'_1)^2+\cdots + (x_n-x'_n)^2}$, and that under this generalization it is still a metric and satisfies ...
2
votes
1answer
119 views

Euclidean Metric satisfying the Triangle Inequality - Is there missing details in the proof given here?

The image below comes from the book Geometry and Topology by Miles Reid and Balazs Szendroi. They prove the Triangle Inequality, which is stated below $(2)$. I am happy with the proof of the ...
2
votes
2answers
76 views

Prove an inequality in $\mathbb{R}$

Let $ p,q \in \mathbb{R}, \; \lambda > 0, p \neq q$ (two points). For the two points $x_+, x_{-}$ with \begin{align*} x_+&=p+\lambda\cdot (q-p)\\ x_{-}&=p-\lambda \cdot (q-p) ...
3
votes
2answers
111 views

Minimal length of non-contractible loops

Not self-intersecting loops on a connected closed orientable smooth surface $S$ must have a minimal length not to disconnect it, e.g. the equators of a torus. "Not to disconnect" is - on such surfaces ...
9
votes
2answers
148 views

A natural-looking distance formula

The distance formula in one dimension is $$D_1 = |x_2-x_1|$$ From the Pythagorean theorem, the distance formula in two dimensions is $$D_2 = \sqrt{|x_2-x_1|^2 + |y_2-y_1|^2}$$ Now, in three ...
4
votes
1answer
522 views

Do projections onto convex sets always decrease distances?

Suppose $(M, d)$ is some $\ell_p$ metric space (not necessarily Euclidean), and $C \subseteq M$ is a closed convex set. Consider the projection function $f_C:M\rightarrow C$ defined such that: ...
7
votes
4answers
425 views

What is straight line?

I have found the definition of line in metric space. It is general but has two problems. Considering about $\mathbb R^2$ equipped rectilinear distance, every line by this definition contains a ...
1
vote
0answers
143 views

Should every line be infinite in both two directions?

Yesterday one my classmate asked me for the the definition of line. I have found it in coordinate geometry, in vector space, and in metric space. The definition in metric space is quite general ...
4
votes
1answer
139 views

Volume and diagonal length of the Hilbert cube

Here's something sort of fun that I gave thought to a while ago, and now that I've done some maturing mathematically I'm curious to see if my musings are legitimate. Let $H=[0,1] \times ...
12
votes
4answers
601 views

Why do we use the Euclidean metric on $\mathbb{R}^2$?

On the train home, I thought I would try to prove $\pi$ is irrational. I needed a definition, so I used: $\pi$ is the area of the unit circle. But what is a circle? A circle is the set of tuples ...
3
votes
1answer
105 views

Discrete Subgroups of $\mbox{Isom}(X)$ and orbits

Let $X$ be a metric space, and let $G$ be a discrete subgroup of $\mbox{Isom}(X)$ in the compact-open topology. Fix $x \in X$. If $X$ is a proper metric space, it's not hard to show using ...
4
votes
1answer
86 views

Challenge question for normed linear spaces.

I have come across the following challenging problem in my analysis course: Let $K$ be a compact convex set in a normed linear space. Suppose that $$\sup_{x,y\in K}\{||x-y||\}=\delta>0.$$ Show ...
-1
votes
1answer
105 views

Taxicab Geometry: solution to $d(P, A) = 2 d(P, B)$ for two points, $A = (4,7)$ and $B = (5,4)$?

I need the solution for $A= (4,7)$, $B= (5,4)$, please also the graph and an explanation about the procedure for getting the graph.
3
votes
1answer
72 views

Compact sets in geodesic metric space

Are compact sets in a geodesic metric space necessarily bounded? What about, if the space is proper?
20
votes
2answers
329 views

A strangely connected subset of $\Bbb R^2$

Let $S\subset{\Bbb R}^2$ (or any metric space, but we'll stick with $\Bbb R^2$) and let $x\in S$. Suppose that all sufficiently small circles centered at $x$ intersect $S$ at exactly $n$ points; if ...
1
vote
1answer
166 views

Maximal color difference

I have a picture consisting of a two-dimensional array of ordered triples (red, green, blue) of real numbers from 0 to 1. I'm looking for something like a norm on pictures which expresses the range of ...
3
votes
2answers
155 views

Given pairwise distances of $N$ data points and find the minimal dimenion of space can fit the data

Given a set $D$ consist of all pairwise distance of $N$ unknown dimension points. e.g. If there is 3 points, ${x_a,x_b,x_c}$ $$D=\{||x_a-x_b||,||x_b-x_c||,||x_a-x_c||\}$$ How can I find the minimum ...
2
votes
0answers
95 views

Inner product and inequalities

Suppose $p:[0,1]\to \mathbb C$ is a curve where $p(t)=u(t)+iv(t)$ and $u,v$ are smooth functions of $t$. Why then is $$\left(\int_0^1 \langle \dot{p},\dot{p}\rangle^{1\over 2} dt\right)^2\le \int_0^1 ...
1
vote
1answer
95 views

Lorentz reflection

What is a Lorentz reflection of $\mathbb R^3$? Is there a way to visualize it? Suppose I have a plane, P, what would (Lorentz) reflecting in it differ from (Euclid) reflecting in it? I know that the ...
1
vote
2answers
87 views

Geometric explanation of the product metric

Can someone describe to me the geometric intuition behind using a mapping $$ ((x_1,y_1),(x_2,y_2)) \mapsto \frac{d_1(x_1,y_1)}{1+d_1(x_1,y_1)} + \frac{1}{2} \frac{d_2(x_2,y_2)}{1+d_2(x_2,y_2)} $$ to ...
4
votes
1answer
175 views

Metric on the unit cube

Let $X$ be $\mathbb{R}^3$ with the sup norm $\|\cdot\|_{\infty}$. Let $Y=\{x\in X: \|x\|_{\infty}=1\}$. For $x,y\in Y,y\neq -x$ define $d(x,y)$ to be the arc length of the path $$Y\cap \{\lambda ...
7
votes
2answers
2k views

Cosine similarity / distance and triangle equation

There is a similarity function particular popular for processing sparse vectors such as textual data (word frequency counts etc.) commonly referred to as cosine similarity. There are two variants to ...
3
votes
2answers
249 views

How to calculate circumference of circle in $L^p$?

I want to find the circumference of a circle as a function of its radius in $\mathbb{R}^2$ with $l_p$ metric. Is the following anywhere near correct for radius $r$? $$ C_p(r) = 4\int_0^r \sqrt{1 + ...
2
votes
1answer
284 views

A property of geodesic metric spaces

Let $(X,d)$ be a uniquely geodesic metric space, i.e. for every $x,y\in X$, there is a unique continuous function $f:I=[0,\ell]\to X$ such that $$f(0)=x,f(\ell)=y, d(f(r),f(s))=|r-s|\ \forall r,s\in ...
2
votes
1answer
121 views

Is there any object/metric within all points are at same distance to each other?

A circle in Euclidean Geometry are all the point at same fixed distance (radius) from a single point called the center. That fixed distance is only from every point to the center, but there are ...
1
vote
1answer
43 views

Distance between $N$ sets of reals of length $N$

Let's say, for the sake of the question, I have 3 sets of real numbers of variate length: ...