All Questions
Tagged with examples mg.metric-geometry
12
questions
1
vote
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99
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Folding a non-rectangular shape into a rectangle of uniform thickness
I think the following might be an interesting subproblem of this question:
Question: For an odd number $n\ge 3$, is there a non-rectangular but still convex shape of area $A=1$, that can be folded (...
- 11.9k
0
votes
0
answers
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Equal volume and projections
Given three unit vectors $u_1,u_2,u_3$ in $\mathbb{R}^3$, can we find some body $K \subset \mathbb{R}^3$ (probably convex) such that the following three things hold
(1) $|P_{u_1^\perp}K|=|P_{u_2^\...
- 233
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8
answers
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When does a metric space have "infinite metric dimension"? (Definition of metric dimension)
Definition 1 A subset $B$ of a metric space $(M,d)$ is called a metric basis for $M$ if and only if $$[\forall b \in B,\,d(x,b)=d(y,b)] \implies x = y \,.$$
Definition 2 A metric space $(M,d)$ has &...
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1
answer
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Worst Case Region for a Convex Hull Heuristic
I am currently implementing a heuristic algorithm for planar convex hulls hand would like to know, for which kind of strictly convex region it exhibits worst performance.
I know that there are many ...
- 12.5k
3
votes
0
answers
266
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Seek "typical examples" for the structure of spaces with two-sided Ricci bounds
By a 1990 paper of Michael Anderson, the following is true:
Theorem. Let the metric space $(X,d,p)$ be a pointed Gromov-Hausdorff limit of a sequence of complete pointed Riemannian manifolds $(M_i,...
- 3,142
12
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8
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Applications of the notion of of Gromov-Hausdorff distance
I am looking for applications of the notion of Gromov-Hausdorff convergence to prove theorems that a priori have nothing to do with it. Examples that I am aware of (thanks to wikipedia and google):
...
17
votes
3
answers
3k
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Nonseparable example in dimension theory?
Could you give me an example of a complete metric space with covering dimension $> n$ all of which closed separable subsets have covering dimension $\le n$?
The question closely related to this ...
- 1,785
4
votes
1
answer
774
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Example in dimension theory
Could you give me an example of a complete metric space wiht covering dimension $> n$ all of which compact subsets have covering dimension $\le n$?
- 1,785
10
votes
2
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Canonical geometric examples
The proofs without words post has some great entries. I'm interested in a similar concept: examples where a problem in math or physics is accompanied by a geometric figure that illuminates some key ...
73
votes
10
answers
10k
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Riemannian surfaces with an explicit distance function?
I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ...
- 106k
2
votes
0
answers
252
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Forgetting extra structure inducing Symmetries
This is a major edit of the original post after receiving helpful comments.
It is often the case when one adds additional structure to make a problem more tractable. When one attempts to forget this ...
7
votes
2
answers
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Example of non-closed convex hull in a CAT(0) space
this is related to this question but is simpler, and hopefully is well-known. There are a number of references that say that the convex hull of a collection of points in a CAT(0) space need not be ...
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