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Results tagged with mg.metric-geometry
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user 1631
Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
1
vote
When shorter means smaller?
An incomplete answer; but perhaps it helps to rephrase the problem as below. The reason the round circle does have this property is that without loss of generality, the map $f$ fixes the origin; and …
- 1,987
-1
votes
When shorter means smaller?
Another tact; consider the set of foldings of our convex shape. Since the fold axis partitions the perimeter into two parts, and by convexity both have finite length, so one of them is at least half …
- 1,987
5
votes
Side-Angle-Side Congruence and the Parallel Postulate
On reflection, SAS tells me that Euclidean geometry has a strong semi-local homogeneity, in that every neighborhood of every point is isotropically isomorphic with some neighborhood of any other point …
- 1,987
8
votes
Accepted
Subset of the plane that intersects every line exactly twice
By AC, choose a cardinal well-ordering of the lines in in the plane and any well-ordering of all the points.
We proceed by transfinite induction.
Suppose $A_l$ is a set of points, no three colinear, …
- 1,987