0
votes
1answer
14 views

The measures used to define Hausdorf dimension versus Haar measure

I don't know if there is a name for the measure, so let me construct it: Given a metric space $X$ we define the outer measures $H_\delta^\alpha$ by $\forall A \subset X$ $H_\delta^\alpha ...
0
votes
1answer
58 views

How to make the Symmetric Distance a metric?

I am trying to construct a family $S$ of measureable subsets or $R^2$, on which the symmetric difference, defined as: $SD(A,B) = Area(A\setminus B \cup B \setminus A)$, is a metric, i.e., different ...
1
vote
2answers
50 views

Looking for a proof that the diameter of the smallest bounding circle is less than or equal to $\frac{2}{\sqrt{3}}$ times the diameter of the set

This came up while I was attempting to solve an old journal problem. It's not the easiest result to search for so I figured I would ask. Let $E$ be a subset of $\mathbb{R}^2$, then the diameter of ...
2
votes
1answer
71 views

Is there any proof for this simple observation? [duplicate]

If we consider the Euclidean space $R^3$, it is simply the space where we live. Here we can find only four point such that distance between any two points is a constant. If we consider the Euclidean ...
0
votes
1answer
27 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} ...
3
votes
1answer
131 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
93 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 ...
1
vote
2answers
70 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
192 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
34 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
98 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
110 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
53 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
66 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
34 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
40 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
173 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
62 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
85 views

Symmetry of a Manhattan Distance

I am having trouble with proving that the Manhattan distance (also known as Taxicab geometry) is a metric by satisfying the condition of symmetry. Can anyone point me in the right direction?
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
220 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
153 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 ...
4
votes
2answers
140 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) = ...
4
votes
5answers
107 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
54 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
67 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
124 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
131 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
127 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
113 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
149 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
550 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: ...
8
votes
4answers
430 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
146 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
140 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
613 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
107 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
87 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
106 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
75 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
331 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
167 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
163 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
98 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
176 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 ...